This may seem like an obvious tip, but it is so crucial that I thought it should be the first place someone would start in improving their FPS skills. For any particular game you play, you should go through the multiplayer maps by yourself to run around and get a feel for the environment. Obviously each map will be different in its features, hiding places, and tricky locations. After you have had a chance to review the map by yourself, look up the map online to see if others have found some spots that you have missed.
I'll include some posts with particular maps and strategies for those maps, so send any requests for a map that you are looking to improve on and we can work through them together.