After you have been working out with weights for a couple weeks, you will want to be doing 3 sets of each exercise. You said your goal is 15 reps. That is a good goal. You will need to add resistance (more weight) to your exercises to the point that you cannot do 3 sets of 15 reps (with perfect form) each. You will be working your muscles to temporary failure. Then as you get stronger, (and you get stronger by healing on your days off) you will be able to perform all three sets of 15 reps. At that point, you add a little more weight, and then start trying to get to 3 sets 15 reps again.

Be sure you know what the proper form is for all of your exercizes. Use lighter weights until you get this down correctly. Do not "cheat" your form just to lift more weight. You will not be exercizing the muscle group that the exercize is targeted for, and you WILL risk injury.

The first few times you work out enough weight to reach your failure point, you will be sore a day or two later. After that, you will have some soreness, but not that bad. (until you add new exercizes)

Remember, your gains are made while your body is healing. Don't exercize the same muscle group two days in a row. I only exercize the same muscle group once a week. (but, then, I'm getting older, and don't heal so quickly.) Twice a week for each muscle group is plenty. If you lift 3 or 4 days a week, split your exercize routine up into smaller routines so you avoid over training. Have a trainer set up routines for different days.

(Very simple example see a trainer)
Day 1 Chest-Shoulders-Triceps-Legs
Day 2 Back-Biceps-Abs


You only make real gains by pushing yourself until you hit your failure point.
If you don't think you are working hard enough, you probably aren't.

Don't worry about free weights yet. Use the machines. They help keep your form correct, isolate the muscle groups you wish to exercize, and will protect you from injury caused by not lifting correctly.