If you want to go through Maine, I would recommend US 201 to QC 173. It significantly cuts down travel time, but if you want to take your time and explore, take I-95 to NB 95, then take the Trans-Canada. In Quebec, it's partially two lane north of the New Brunswick border, but most of this stage is freeway.
In Montreal, take exit the Trans-Canada (A-20) to Autoroute 30 south of the city if you want to avoid traffic. It does have a toll though on the western side of the freeway. West of Montreal, you have to decide if you want to go through southern Ontario (ON 401) or northern Ontario (ON 417). The north is a lot more rural, but has more two-lane highway. It depends on what you're looking for.
If you take the southern route, follow A-20 to ON 401. ON 401 goes directly to Toronto. Traffic in Toronto (like any other major city) is pretty bad though, so I would avoid it unless there's something there you want to do (like driving the busiest highway in North America). You can avoid Toronto by taking Brock Road in Pickering to ON 407 north of the city. There's a hefty electronic toll though. Afterwards, take ON 400 to ON 69 towards Sudbury.
If you take the northern route, you basically follow A-40 to ON 417 through Ottawa, the nation's capital. West of there, the (mostly) freeway trip ends, and it becomes two-lane roadway. ON 17 I believe is the longest highway in Ontario. It's pretty scenic, but gorgeous in the summer.
Regardless if you take the northern or southern route, you will end up on ON 17 in Sudbury. You can take ON 11 north of the city, but it's much longer, and more scenic (and probably less traffic) than ON 17.
ON 11 and ON 17 meet up in Nipigon, which is the only way west. The Ontario Government is building a new bridge on the Nipigon River, so there may be traffic delays. Follow ON 17 to the Manitoba border. Once you enter Manitoba, stay on MB 1 until you hit the Perimeter Hwy (a ring road surrounding Winnipeg). You can choose to go south, through, or north of the city. The main Trans-Canada route is through the city.
The official bypass is MB 100 which goes to the south of the city. However, I recommend going through the northern part (MB 101). It has fewer traffic lights and less truck traffic. Either way, the ring road will meet up again at Highway 1 west of the city.
There's not really much else to say about the Prairies. It's divided highway all the way to the British Columbia border. Calgary has another ring road (AB 201) that you can use to bypass the city. Once again, the Trans-Canada goes straight through it. West of Calgary, the Trans-Canada becomes freeway, and you will see breathtaking images of the Rocky mountains.
In BC, Highway 1 becomes two-lane again for most of its length. West of Kamloops, it becomes freeway. Take BC 5 (Coquihalla) just like Jakeroot suggested. If you want to avoid Vancouver, you can take BC 17 to BC 91. BC 91 southbound will lead to BC 99. BC 99 leads to I-5 in Washington state.
Obviously, I don't know any tourist attractions / road related goals you would like to strive for, but this is a basic guide. Quite the road trip. Here's the map:
https://goo.gl/maps/iw7FB9XMkR52
