You are generally assigned to the best aircraft your seniority can hold. Usually that will be the most junior equipment the airline operates.
A lot can depend on need. For example, several years ago at Mesa, new-hires went to the Beech 1900. However, if FOs were needed on the next higher aircraft, the Brasilia, new-hires could be assigned to that airplane. Then, I heard that last summer Mesa was putting new-hires directly into the RJ - I don't remember if it was the ERJ or CRJ.
As you build seniority, you can bid more senior equipment or crew positions. It may take years to move up the chain. Then, you may find you won't want to bid more senior equipment because the line you would hold on that aircraft will be junior to pilots already assigned to it and may not be as good as the line you were holding as a senior pilot on junior equipment. In other words, taking the Mesa example, you might have been Pilot No. 1 on the 1900 and had your choice of schedules. You bid the Brasilia and you drop to the bottom of the totem pole and have to take what's given you until work your way up on that aircraft. Much of that may also depend on how your company handles seniority.
I knew a Delta pilot who was FO on the L-1011. He was based in LAX and didn't have to work that many days per month. He most certainly was qualified to upgrade. He held Otter and multiple jet type ratings and was a DE for these jets. He lived on the West Coast and had a great second business being a DE and running his own 141 type school. He told me that if he bid for Captain it would be on the DC-9, which was junior equipment, and he'd be domiciled in Cincinnati. That would have meant a commute for him. His seniority obviously let him hold FO on the 1011 and he was happy with it.
The bottom line is if you are senior you can about write your own ticket. If you're at the bottom you have to take what's given you.