I don't know about all states but in Virginia you can pay a fee and drive without insurance. Not a good idea but its legal and that is why most folks carry a rider on their policy to cover them if they are hit by an uninsured driver.

I had to pay extra costs when my children began to drive (boys under 25 are more expensive) and I think most insurance companies are like that. The cost depends on the vehicle, sex of the driver, type of coverage and several other factors (my company offered a "good student" discount and a discount if they had received "driver education" at school.

There are no restrictions on the type of private car (including some trucks - you can rent a rather large van and drive legally for moving) you can drive but you have to have a special license for large commercial vehicles, motorcycles and vehicles for hire that carry passengers.