You will likely get your first job with a commuter airline. From then you can work your way up to something as big as a 777. And yes, you do only get trained on one plane in a comapny unless you get switched. Based on the pilots I know, you get put on whatever plane they hire you for. Look for an airline that flies a lot of 777's to make it more likely. But if you are not a commercial pilot yet, that may be a long way off.
It depends on what airlines are hiring at that time, and what types of airplanes they use. After you have the job, it becomes a matter of seniority. You take the routes they give you, and fly what they are using for those routes. That is a management decision made 'way upstairs somewhere.
They will expect you to qualify in whatever they need you to fly, and you may be sure that if you stay in the business, you will fly at least several different types in the course of a career. The more seniority you build up with a company, the more control you will have over where and when you go and what kind of equipment you sit in.
It all comes down to seniority. The guys who have worked for the company longest fly the trips they want and the new guys take what ever is left over. Probably your first 10 years you will fly junk air planes into the crummiest airports in the country. As you gain more seniority you can decide what trip you want. I think the airline decides what aircraft to give you depending on the destination and the number of passengers. One suggestion is do your research before you decide to get into this profession. I wanted to get into it for while but heard a lot of bad things about it, and there are tons of misconceptions.
Whatever your juniority will hold. It's all based upon where you are on the list. EVERYTHING from seat, equipment, domicile, vacations, trips, and furloughs is dependent upon your seniority. Never turn down a class date or an upgrade if your number can hold it.
When you are initially hired you may or may not be given the option to give a preference as to what route or aircraft you want to operate. Based on the company's needs, they may or may not give you your requested aircraft. Given enough time with a company you will gain seniority and be able to bid on the position you want when one opens up.
Most airlines that operate routes with 777 will start you out on their smaller routes which more than likely utilize the 737. You are then sent to training to get "type rated" in the 737. Some companies cover the cost of training you, others hire those who only already have the proper rating. Should a position open up on a 777 route and you are selected to fill it, the company will then send you for training on the 777 aircraft to gain a type rating for the relevant aircraft.
So it is a matter of working your way up the ranks in a major airline, as most 777 routes are held by very senior pilots.
Pilots are not limited to only one type rating.
By the time you qualify there may be many fast rail services so your career options will be limited.
The 777 will be retired way before you are ready so be more flexible.