How many sets should I do?

Instead of working out your whole body every other day... it makes much more sense to work on specific parts of the body. Common splits include:

1. Upper body - lower body (Upper body Monday, lower body Tuesday, hybrid of both friday)

2. Triceps/pecs Monday, biceps/back/shoulders Wednesday, Quads/hamstrings/calves Friday.

This way you can focus more on each body part and help it grow and get stronger faster. Instead of working out 10 different muscle groups lightly each workout, it is much better to work out 1-3 muscle groups to exhaustion.

3 sets of 8-12 reps is a good reference point for weight lifting. Generally, to become stronger you want to lift heavier with lower repetitions, and to become more "cut" you want to go lighter with more reps. It all depends on what you want :). 6-12 reps is the happy medium, where you gain size and strength.

EDIT:

Six exercises is good, if you have enough time... typically I can get in 3-4 exercises in an hour (squats, leg press, calf raises, good mornings/deadlifts) and that's all I have time for. If you have more time, the better it is, so go for it!

Yes, three sets of each muscle group is good enough. Reps depends on how you want to train. If you want to be lean and cut, then do light weight with high reps such as 8-12. If you want to be strong, then 4-6 reps of heavy weights is good.