I've lived in Québec, Montréal and Gatineau (where I currently call home) and each is a great place to live for different reasons. I can't really comment on Trois-Rivières, although I have been there many times.
1.) Quebec City is probably my least favourite of the three (not that it's a bad place). You can really divide the city into two parts, the richer more affluent area on top of the hill (Montcalm, Sillery, Ste-foy, Cap-rouge, etc).. and the poorer parts of the city, which are at the bottom of the hill (Charlesbourg, Vanier, Beauport, St-Roch, etc -- with the exception of Lebourgneuf, especially le Mesnil, which is fairly affluent / upper-middle class). Nobody really lives in the Old-Quebec, as services there are few and far between, ie no grocery store, etc. (I made that mistake for a year, oops). Your experience will really vary depending on which of these two regions you live in. People on the hill are pleasant, but also kind of snobby/aloof. They'll point out your accent (if you have one) and correct you when you make mistakes in French (passive aggressively). There's also a tendency for people here to go out of their way to use the correct french words -- avoiding english words at all cost lol ("windshield" is "pare-brise", etc). There are a lot of cute areas up on the hill (Ave McGuire for example). At the bottom of the hill, it's more working class/working-poor. It's not dangerous there, but people can be a little more aggressive/confrontational. Unless you're familiar with typical quebecois culture, this might put you off. That being said, if you're looking for a more authentic Quebecois experience, I would go in these areas.
Unless you live along a Métrobus line, public transportation is nightmare in Quebec City. Traffic isn't too bad, but I find drivers there to be much more aggressive than in Mtl. There's an extensive, if confusing highway system in QC which makes it easy to get around; but personally, I prefer driving in Montreal (which might make me an anomaly).
It's really hot in the Summer (it's the pool capital of North America I believe) and really cold, with a lot more snow in the winter.
The city is pretty affordable, especially when compared to other cities in the country.
Nature isn't that far away (if that's important to you)... but not, I'm going to ride my bike there, accessible. You're far from bigger cities.
2.) Montreal is really just a fun city... You can get by in English if you want (depending on where you live). The further east you go, the more (working-class) french the city becomes. The transition from English to French happens somewhere between St-Laurent and St-Denis. But most people in the downtown core are bilingual, and can converse in either language.
There's really no shortage of stuff to do in Montreal (in either English or French)... and there's no way I could list them all here. Part of what makes it interesting is that it's filled to the brim with students. While this often contributes positively to the city's vibe (especially where arts are concerned), it can also make the city overtly-political sometimes, which may turn some people off (see anti-police brutality march, mccill's social justice warriors, nationalists from UQAM, etc). Students in Montreal love to protest.
The city is also very multi-cultural, which makes it's restaurant scene pretty amazing. You could probably visit a different restaurant every day, for the rest of your life, and never have to eat the same place twice. I'm not even sure if that's an exaggeration. lol. There's no shortage of good food in Mtl.
Public Transportation, if you're on a metro line, it's pretty good. Even the bus system is pretty good (imo). You don't really need a car in Montreal, unless you want to go to Ikea or leave the city.. and for that there's car sharing. Driving in Mtl can be a nightmare, because there is always construction... and you have to follow detour signs which often lead you nowhere. It can be infuriating.
Weather wise, it's a lot like Quebec city, perhaps milder?
With the exception of the mountain in the middle, there isn't a whole lot in terms of nature in Montreal. The city is very dense, and the island is almost completely developed (less so on the western-end of the island). You'd need to drive north towards the Laurentides to get back to nature. That being said, there are smaller, forested urban parks all over the place.
Montreal used to be crazy affordable. My first 2 bedroom apartment cost me only 350$/month.. it's gotten more expensive lately.. but nothing too crazy.
3.) Gatineau is really just a suburb of Ottawa, but on the Quebec side of the border. That being said, you can pretty much find everything you'll ever need on the Gatineau side (and what it doesn't have, Ottawa probably does). Gatineau, like Montreal, is very bilingual. The further west you go, the more english it becomes (Aylmer)... There are many cute areas on the Gatineau side as well.. Chelsea & Wakefield come to mind, both of which have a small quaint village feel to it (not what I would call Quebecois though). Hull & Gatineau are more French, and are both working class for the most part. Consequently, the French spoken here is perhaps not as refined as you'd find in Quebec City or in the Frencher parts of Montreal. People are friendly, but can be a little trashy in some places (pointe gatineau, hull).
What makes Gatineau such a great place (when compared to the other two cities) is its proximity to nature. The entire city is essentially cut in half by Gatineau park (which I'm pretty sure is bigger than the city itself). It also has an extensive network of forested trails along the ottawa river. There are a ton of hiking trails (which, let's be fair, all resemble each other) all over the city. We're not talking the Rockies here, in terms of beauty... but if you like sports that take place in nature (cycling, walking, skiing, rollerblading, long boarding, you couldn't ask for better). Go 20 minutes north, and you're in Val-Des-Monts.. which is littered with lakes (and cottages). Again, if nature is your thing, this is a great place to go.. canoeing, kayaking, etc..
The city has only two freeways.. and they can get pretty congested during rush hour (especially the 50) if you end up working in Ottawa, expect to spend at least an 45mins in your car. The bus system isn't exactly great (the worst between the three), but if you live along the Rapibus lines, it's more than adequate.
One thing to keep in mind.. the health care situation in Gatineau is atrocious.. It's the literal embodiment of what pro-market Americans fear of when they think of socialized medicine. Gatineau hospital I'm pretty sure was ranked amongst the worst hospitals in the western world. Most people I know.. usually just go to Ottawa.
Weather is similar two the other two, but milder.
Also, another thing to keep in mind is that Ottawa/Gatineau is between Montreal (2hrs) and Toronto (4hrs).
Gatineau is also very affordable.
Anyway.. to recap.. if learning French is your goal, I'd say Trois-Rivieres (for typical Quebecois french) and Quebec City (more formal/international french) are your best bets. If you want to experience great culture/food while being able to live in both languages, while having access to the best public transportation system in province, then I'd say go to Montreal... but don't go there to learn French. If proximity to nature is important to you, learning French is not your primary goal, and being able to live almost completely in English (while having access to French speakers), i would say Gatineau is your best bet.
That being said, they're all great cities.