Three things immediately come to mind. Bad new bulb. Check the bulb somewhere else. Bad switch. Usually, they just don't feel right. Turn off the power. Carefully turn the power back on to check it with a meter. Bad socket. If you have power to the socket and a good bulb, the socket is probably bad.
If you don't feel comfortable turning the power on at the breaker box with the switch or sockets exposed, don't do it. Call someone that does.