What exercises to do
You want to choose about 8-10 exercises, which comes out to about one exercise per muscle group. The list below offers some examples. Ask a trainer to critique your form until you get it right. Work each muscle group 2-3 times a week.

Chest: bench press, chest press machine, pushups, pec deck machine

Back: one-armed row, seated row machine, back extensions, lat pulldowns

Shoulders: overhead press, lateral raise, front raise

Biceps: bicep curls, hammer curls, concentration curls

Triceps: tricep extensions, dips, kickbacks

Quadriceps: Squats, lunges, leg extension and leg press machines

Hamstrings: deadlifts, lunges, leg curl machine

Abs: crunches, reverse crunches, oblique twists, pelvic tilts

How many sets and repetitions to do
(For example, if you lift a weight ten times, rest a minute then lift it another ten times, you just did two sets of ten reps)

For fat loss: 1-3 sets of 10-12 reps using enough weight that you can ONLY complete the desired reps.

To gain muscle: 3+ sets of 6-8 reps to fatigue. For beginners, give yourself several weeks of conditioning before going to this level. You may need a spotter for many exercises.

For health and endurance: 1-3 sets of 12-16 reps using enough weight that you can ONLY complete the desired reps.

Figuring out how much weight to use for each exercise
Write it down once you figure it out. This helps next time, and you can track your progress as you get stronger.

Pick up a light weight and do a warm up set of the exercise of your choice, aiming for about 10 to 16 repetitions.
For set 2, increase your weight by 5 or more pounds and perform your goal number of repetitions. If you can do more than your desired number of reps, heavy up again for your 3rd set.
In general, you should be lifting enough weight that you can ONLY do the desired reps. You should be struggling by the last rep, but still able to finish it with good form.