Could be the ballast. Can get them at any home hardware store. Some basic electrical skills needed (Power - OFF, Disconnect & connect wires).

You can't use just any fluorescent tube (bulb) in some fixtures. Some fixtures require a particular type. It should tell somewhere on the fixture. That might be the problem.

Turn off power (duh) and open the fixture. There is probably a black brick of a ballast on the inside with two red and two blue wires, a black, white and green. Cut the wires so you have some about 4" wire on the ballast, then take ballast to Home Depot. Have them match the ballast, then rewire into the fixture. Not too complex, the ballast has a wiring diagram on the front of it, and it is easy to follow. Don't forget the small wire nuts you will need.