Here's some tips for long trips ranging from a few hundred miles to several thousand (I've taken a 10,473 mile trip once:
http://www.sykotyk.com/supertrip ). I bought a new car at the end of July 2015 and already had 19,000 miles on it by the end of the year.
First, the older the vehicle the more likely something will go wrong. That should be obvious. Even if everything was in tip-top shape leaving home, extended travel can cause wear. And if things start going bad, they can snowball. For instance, a CV joint (Constant Velocity, the shaft coming from the transmission to your front tires on a front-wheel drive vehicle) can go from fine to 'growling heavily on turns' in no time. Also, wheel bearings and ball joints can go bad quickly, as well. These are some of the 'to watch for' issues with cars, especially older cars.
I've found cars tend to wear better under constant, steady use rather than intermittent travel. Sitting a car in the driveway only driving a few miles a day and then expecting it to drive 3,000 miles in a week is like you sitting on a couch for several weeks and then expected to run a marathon. It's a lot more difficult.
So, for noises: if you hear a growling sound when turning at slow speeds, it's your CV joint. A light growl might just be some contaminant (the sleeve ruptured and dirt/water got into it, sometimes you can just repack with grease and a new sleeve). But, it's usually a sign of wear among the sprocket and the ball bearings inside of it. The worse it gets, the more 'chatter' you'll hear at low speeds. The sound of the road and engine make it almost impossible to hear at higher RPMs and it's most noticeable at slow speeds making sharp turns. This is a problem that generally gets spotted well before it becomes critical. But, if you start to hear it, be aware of what it means.
Second, is growling from the wheel that picks up pitch as you speed up. Generally, this is the wheel bearing. It's known for not going away no matter what you do. Though sometimes turning the steering wheel a little left or right may relieve some of the pressure against the two parts of the bearing that are now touching. This is bad. I drove to Tennessee with a wheel bearing that had just started making noise and by the time I had it home it needed replaced. I risked it because I know the longevity of a problem compared to the need to fix it. If you're not that clearcut on what a car does, you shouldn't test your luck. A wheel bearing failing is going to require a tow, a shop bill, and some downtime. And it can cause other damage, such as to the CV joint/half-shaft.
Next up, is the ball joint. This is the pin sticking through the bottom of your wheel assembly (the A shaped bar sticking out from your frame rail). It's actually attached to the end of your strut assembly and is a 'joystick' like bearing. It allows the strut to move up and down with the road bumps and for the tire to pivot around and stay all connected to the strut and the frame. A failing ball joint is signified, at the beginning, by a slight 'tapping' sound when you hit a bump. The metal is wearing, and there's now slight space between the metal pieces of the bearing. A strong jolt smacks them together. As wear accumulates, you'll feel your tire riding rough or 'sloppy' on the road, and the tire will begin to wear funny (cupping, uneven, wavy). A failed ball joint is catastrophic and will leave you on the side of the road if not in the ditch or a bad accident. If it separates, you have lost that wheel. Ever see a car on the side of the road with a front wheel wedged up under the car or bent at an odd angle? Good chance it was the ball joint. The wheel will do what it wants after the ball joint goes as it lost it's pivot point.
Also, there is the tie-rod end. This is the rod sticking out from the steering mechanism (usually rack & pinion controlled by the power steering pump). Each side goes to each tire and includes a pin sticking down through the end into the wheel assembly. It turns the wheel. The bottom has a castle nut (looks like a rook in Chess) with a cotter pin through it and bent over (this goes through the teeth of the castle nut to lock it in place from loosening. It's bent over to keep it from falling out. Always inspect this castle nut and the one on the ball joint. If either cotter pin looks broken or missing, replace it immediately.
My experience with a tie-rod end is that they tend to 'freeze' from lack of grease (there's a big grease fitting on top, fill it full of grease until you see it start to ooze out of under the rubber ring around the gap). Some cars are 'sealed' with no fitting, these are just waiting to go bad and require more work and money. If they're not greased, they will start to touch metal to metal without something protecting them. The constant turning will grind down the shank of the pin or the ring it goes through, meaning the pin gets smaller compared to the opening. If it weakens enough, it can snap. And now you have a free wheel uncontrolled by the steering wheel. Which is absurdly dangerous.
Luckily, two of these things are generally fitted with grease fittings (newer cars, sometimes don't). But, if you see grease oozing from one of these spots, it's a problem. A big big problem. The other (the CV joint) can start to leak, usually from a crack in the accordion-like sleeve that goes around the joint. When you look in at it, wipe it clean of dirt and see if there's any grease between the rubber/plastic indentations. If there are, the boot needs replaced, at minimum.
Luckily, these aren't that expensive to fix, even at a service shop. If you don't know what you're doing, they can be a pain to do it yourself (plus you would need to run to autozone to rent the tools to split the ball joint or tie-rod end so you can replace it. Also, if you do either, you will need an alignment. If you're doing it at a shop, they'll probably require you get an alignment. This isn't an upsell, but a basic requirement whenever you mess with the alignment of the front axle.
I figured brakes and rotors are 'basic' maintenance. Most brakes today have squealers on them to let you know you've reached the minimum thickness generally allowed. There's also usually a notch in the pad to show you where it is as well. Once you've reached that depth, you replace them. One good tip, for your rear/emergency brakes is to use your e-brake when you park from time to time. And especially when parking on hills. This regular use keeps that brake working. Lack of use can cause it to seize up. It's also a good idea to test your rear/emergency brakes every once in a while. Best/easiest is to park, and while you're in gear and have your foot on the brake put on the e-brake. And then let go of your service brake. The car should bite into the brake and slouch a bit and should not go anywhere. If it does, the e-brakes are bad or the mechanism to activate them failed. These brakes generally aren't used during every day driving. They're entirely a back up system. Which is why you must test them from time to time that they work.
It's also good when parking on hills to activate your e-brake before putting the vehicle in park. Set the e-brake and then put the vehicle in neutral and let it 'rock' backward or forward where the e-brake has it locked in place. Then, put the vehicle into park. This helps protect the parking gear, which is usually a pin or plate the gear stops against that keeps the gears and the connected wheels from rotating. This also makes taking it out of park on a hill much smooth and safer. And, it never hurts to turn the wheel into the curb when parked on a hill if you don't already know it (if on the right side of the road, turn the wheel to the right, so in case the parking gear and/or e-brake fails, your car will use the natural friction against the curb to stop it from rolling away. (there's a lot of people that don't seem to understand this very simple concept).
Now, back to long-distance driving. One thing to keep in mind is oil consumption. Even on a brand new car it is not unusual for a car to 'burn' through oil on a long, extended trip where you might be driving 2000 miles in just a few days. On short trips, the car's oil may heat up but it's viscosity keeps it working. When the engine cools, it thickens back up again. On a long drive, the engine does not get the chance to do this. The oil gets thin and stays thin and the longer it runs like this, the higher the likelihood the oil breaks down and you start to burn it off.
On a 3,000 mile trip (on fresh oil from the start) it wouldn't be absurd to lose a quart of oil during that time if you did 3,000 miles in 4 or 5 days. The recommendation would be: use thicker oil. Especially if you're heading further south into the southeast summer, or maybe the mountains where your engine will work a bit harder). If you usually use 10w30, maybe use 5w30 or straight 30w. Already use 5w30 (like a lot of cars recommend, mostly due to the American Southwest and humid southeast), maybe run 15w40 in an older engine. Takes longer to break down. Just don't use thicker oils in winter. Could cause excess engine wear. Especially if you're going from down south to up north in winter. Other things could be STP Oil Treatment, Lucas Oil, etc, things that help thicken the oil and make it stick to the moving parts a bit easier. I use lucas oil in all my old cars, and it really helps boost the oil pressure back to the 'like new' pressure when the vehicle was younger. Also, it helps keep the engine operating a little cooler as there's less friction, which equals less heat.
On very long trips, you probably will notice your MPG will drop. For instance, my brand new car gets 39mpg around home on the freeway and highway for short trips and about 31-32mpg in town. But, on a long trip with mostly highway miles, I'll get around 36-37mpg at absolute best. And that's working to try and milk it for all its worth. Longer a little gas engine works, the less efficient it becomes. It's not like a diesel where it's opposite. A diesel may be inconsistently low MPG at first, but you drive one all day, and the MPG will be steadily higher the hotter the engine gets and stays.
Fuel pump/filter, are generally maintenance scheduled items. if your pump fails, you're probably not moving. Because by the time you realize it is failing, it's because it's not giving enough gas pressure through the line to the plenum/intake. To test this, there is a small 'valve stem' like pitcock located somewhere near the plenum where the fuel line comes into it, or near the intake or throttle body. You simply stick a pin or nail or something into it, there should be gas shooting out of it when you push down. If there isn't, you have no pressure which usually means the pump failed. Turn the key on, but don't start the engine. You should hear the pump start up from the rear of the car. if it doesn't, it's the pump. If it does, it might not be working fully, or there's a blockage with the fuel filter. The fuel filter is sometimes located inside the tank with the fuel pump but sometimes located along the fuel line going to the front of the car. Usually not far from the tank, though. My old Chevy Celebrity was accessible from the wheel well, and was just a few bucks to buy a new one, and two plastic clips connected it to the fuel line. An easy 2-minute job once the wheel was off.
Next up: coolant system. If your car starts overheating, one tip is to loosen the radiator cap. AFTER YOU LET THE CAR COOL OFF. Just turn it a quarter or half turn. This helps relieve the pressure in the system. If the car wants to overheat again, coolant will overflow the radiator here, but it will keep the car's temp from skyrocketing. A lot of times, the car overheating is a symptom of the thermostat going. But, first is to listen for the fan or watch to see the fan spin up when it reaches around 210F. In my old celebrity, I simply removed the thermostat. The general purpose is to keep the coolant from flowing until the engine was hot enough to need to be cooled. Usually 195F or so. If the temp were below that, the thermostat would stay closed. if it goes above that, it opens to let the water circulate in the system. Once at 210 or so, is when the fan would then also kick on. The thermostat also controls letting warm coolant through the heater core to allow you warm air through your vents. It should open the same time the engine reaches optimum temperature and the thermostat opens. With mine removed, my car starts blowing warm air almost instantly, though it isn't that hot but it builds up slowly. Newer cars have enough sensors that a missing thermostat probably sends the computer into convulsions. Most cars, the thermostat is located near the plenum or intake along the end of the engine on the left or right side of the car (for sidemounted front-wheel drive). It's a 'gooseneck' and the coolant line runs off of it to the radiator (top hose of the radiator, follow it to the engine). Usually two or three bolts holding it in. There should be a fiber gasket between the gooseneck and the head, and the thermostate is usually in a rubber ring in a notch in the head or gooseneck. Carry a tube of RTV silicone (hightemp red). Take the rubber ring and thermostat out, use a razer blade (carry a utility knife, they come in handy) and scrape the rest of the gasket and debris from the head and the gooseneck. Take some of the silicone, spread it on the gooseneck, and put the bolts through and line it up, and then push it on and tighten the bolts. The car will run without a thermostat. If you're on a trip, this a simple fix for a potential big problem that can at least get you home. Especially if it's an older vehicle.
Anyways, there's always a lot more. But, those are the things I worry about most when taking long trips. When you work on cars, especially your own, you start to pick up on how they act and what sounds they make. Just keep an ear for when the sound changes. Keep an eye on MPG as that is an indicator of a problem. Watch for grease and keep your car clean. When you go through a car wash, spray around through the rims and the wheel housing to clear away all that road grime and get a clearer view of everything when you do take a wheel off.