To improve your game in general core stability is key in tennis, ask your fitness instructor to write you a programme based on increasing core strength. This will mostly benefit your serve of course, as all of the power comes from your core.
More specific exercises to do in the gym are things like shoulder turns to improve your ground strokes and lunges which will help you retrieve shots.
Obviously don't neglect your cardio though, speed and stamina are vital if you're a serious player - do both sprint drills and regular 5-10k runs.
Finally, having a good stretch routine between work outs will improve your flexibility which you need in abundance on a tennis court with all the twisting and turning you do in every match.
It takes a lot of hard work and time off the court to become a top player. Good luck
There goes the terribly short posts by Bernard.
Just to name a few excersizes.. be patient though, I don't know their proper names
1. While pointing your elbows in the air, lift dumbells by extend your forearm up into the air. This helps tremendously with increasing your racket head speed when you snap your arm for a serve generating pace obviously.
2. Any ball tossing excersizes that work your core is great.
3. Lunges and any excersize that works your thighs plus knees are very key because your legs take a beating in tennis.
4. Those one excersizes where you pull a string that holds weight, uh I have absolutely no idea what they are called. But the motion looks like you are doing one of those "arm sawing" celebrations like Stepanek does.
5. Any work out really helps because tennis uses many muscle groups, and even having some strength in those areas where it doesn't seem necessary, such as your pecs, do help in some situations, especially those high balls.
6. Run. Being in tremendous running shape helps SO much. You heal quicker after long matches, you can practice longer, body feels younger and you just overall become faster on court.