I am trapped in a contract with a phone that doesn't support many apps (Nokia C3-01). If I buy a smart phone of some kind, can I swap the SIM over and use it as a smart phone without problems?
As long as the phone you buy is not locked to a network (or an iPhone) then yes.
I swap my sim between my Samsung Galaxy smartphone and a cheap and cheerful Panasonic (better battery life and absolutely not desirable to thieves) depending on where I'm going to be.
Very likely. You'd probably find on first use your service provider picks up on the change of handset and sends a handset setting text to make sure it works fine on their network.
Be aware that if you have used handset memory to store information on your old handset there may be a bit of fiddling about to transfer it to the new. If you can copy any handset stored data to the SIM before moving it across, that's the easiest way to go.