The carrier trains you for their aircraft. If you don't already have a type rating for your new aircraft, then you'll get one as part of the checkride at the completion of Captain training. (Maybe you flew that aircraft at another carrier, or you were an International Relief Office -- a type-rated F.O., or maybe you simply got the type rating as part of F.O. training on that aircraft earlier.) If the carrier needs you to have the type rating on your new aircraft as an FO, you'll get that as part of your training for that aircraft. If you don't need it, you probably won't.
Once you are in the system, everything is driven by seniority -- what do you have the seniority to hold -- and the rules of the contract.
Each fleet at each carrier, and more often than not, each domicile at each carrier, will have different dynamics. In some cases, pilots "camp out" in one aircraft at a particular base for a long time; in other cases, after a couple of years, they're typically bidding (exercising seniority) to go back to training for the next step up the ladder (MD80 to 737 to 757 to 777 FO, then MD80 Captain to 737 to 757 to 777). Different contracts have different rules; some allow a pilot to camp out as long as he wants -- perhaps he's enjoying the quality of life being a senior FO on a nice aircraft at a base close to home, and he isn't interested in upgrading to a bigger aircraft (where he'd be less senior) or the left seat of a smaller aircraft (where he'd also be rather junior) at a base far away. Other contracts (historically) have had provisions that once you can hold either your same seat on a bigger aircraft (or the left seat on any aircraft for an FO), your clock starts & within so many months, you have to upgrade (assuming one is still available) -- for whatever reason that contract was written to avoid having lots of pilots camping out in lieu of upgrading. All very dependent on the contract in place where you work.
Also, the state of the economy and of the airline can drive a lot of this... with Delta retiring many, many senior captains over the next few years, there are lots of opportunities for progression. When everything tanked after 9/11, forward progress stopped and in many cases people moved backwards... MD80 captains went back to being FO's, widebody FO's went back to being narrowbody FO's, junior FO's were furloughed, etc. During the stagnation of the 2008+ recession coupled with the change to Age 65 retirements, nobody upgraded due to retirements (okay, very few retirements), and almost nobody was growing their fleets, so pilots stayed put much longer than they might have otherwise -- there was nothing available to bid FOR. You can only exercise your seniority for a slot that is open.
I'm also wondering how you get selected for the aircraft you will fly once reaching a major
It's essentially whatever aircraft/base needs the bodies that month. One class may get a lot of 737's and a few A320's, another class gets all 767's. Often, the vacancies in the 767 would be filled by pilots moving up from the smaller aircraft, but that doesn't always occur, and it's not at all unheard of for newhire FO's to sometimes go to much larger (read, better-paying) aircraft. Of course, they'll be junior on that aircraft for a long time; they'd get a "more senior" schedule -- fewer weekends & holidays at work -- sooner on the smaller, more junior aircraft.
Likewise the domicile -- if JFK has the openings, you go to JFK (I'm so sorry). If JFK is full & ATL has the openings, you go to ATL (which is probably better, unless you live close to New York). Easier to change domicile than to change aircraft, since no retraining is typically required. (Maybe an "international" course if your old domicile flew only domestic and your new one flies international -- but that's no big deal.)
Welcome to the airline industry!