I would also recommend handling the gun a lot (unloaded, of course). Always make sure it is unloaded, each time you pick it up, even if you haven't bought ammo yet, even if you put it down just a minute ago. It's a good habit to get into - vital, really.
As long as you don't flip the cylinder shut like they used to do in gangster movies, there is little you can do to harm it. Dry fire it a lot, both double-action (by just pulling the trigger) and single action (by cocking the hammer first.) Dry firing even thousands of times will not harm this gun - if it does, something was defective! Become intimately familiar with its feel. Try handling it and dry firing it with your eyes closed. When you pull the trigger, try to move your trigger finger and nothing else. When you dry fire with your eyes open, focus on the front sight. It should not move when you pull the trigger (or at least very little).
Learn how to de-cock the gun after you've cocked the hammer. That will be something that you will undoubtedly need to do at some point, and it's best to learn it at home with an unloaded gun than at the range with a live round under the hammer. (Briefly, to uncock a revolver hold the hammer back with your off hand and pull the trigger. Then keeping control of the hammer, ease it down to rest against the frame, releasing the trigger after the hammer starts to move.)
When you do go to the range with your gun, go with someone who is familiar with firearms and can help you out with regard to stance, sight picture, etc. Under ideal circumstances, go with someone who owns a S&W .22 revolver so that you can start with that instead of your .38. It's much easier to learn on a .22 than anything else because recoil and muzzle blast are negligible (with proper ear/eye protection of course).
First time, don't even worry about accuracy. Turn your target around so that it's just a blank sheet and just aim to the middle of it. Don't worry about where the holes appear. Load a single round and without looking where it's at, close the cylinder - like Russian roulette, with a target. Aim at the blank paper and focus on the front sight. When you pull the trigger try to move only your trigger finger so that when the hammer falls on the empty chambers, the gun should not move at all. When the hammer falls on the live round it should be a complete surprise. This is a good way to detect whether you're "flinching" or not. Once you've graduated to fully loading the gun, leave one round out every now and then as a check.
That'll get you started. By the way, shooting is a lot of fun!