I've been playing tennis since I was about 8, so that's quite a while lol. As one person has said already, it is always a good idea to get back to the centre of the court between shots - this way it is easier for you to get to the next shot since you can cover all the court quickly.
Something which comes with practise is being able to predict where the ball will go once the opponent has hit it. This is a useful skill to develop as you will expend less energy running all over the court and instead be able to return it much faster. As I say, this comes with practise, so play more tennis and you'll get better at it.
When serving, there is no correct method - all pros do it differently. However, some general rules are:
1) Keep your weight on your front foot which should be your non-racket foot (i.e. if you are right handed then your left foot should be forward and vice-versa);
2) Throw the ball as high up as you can since this will give you a steeper angle over the net, making it harder for the opponent to return;
3) Hold the handle of the racket as though you are shaking hands with it. The racket head should be perpendicular to the floor (i.e. if the floor is like this ___ then your racket head should be like this l) and the handle pointing towards you. Put your racket hand next to it and grip with your 4 fingers first on the bottom, then wrap your thumb over the top. Sorry if this bit didn't come across too well - it's very difficult to describe when you can't see what I'm on about lol!
4) Always keep your feet behind the base line (the back line of the court) and serve to the service box (the boxes on the court close to the net) diagonal from you.
Hope this helps, if I think of anything else I'll post it