Water conducts heat away from your body 20 times faster than air, so the only thing keeping you from freezing after 15 minutes of diving is your wetsuit. Excellent care is a must to keep it insulating well for a long time.
After a dive
Most people know this one. After a dive, rinse the wetsuit thoroughly in fresh water. Take a moment and run your hands over it, rubbing away any salt that may try to dry inside. This basic step will do more for your wetsuit than anything else. Don't forget to do the same after pool dives! Chlorine kills everything---that's why they use it in pools.
Also, let the wetsuit dry inside out.
After a trip
Every once in a while it's a good idea to wash your wetsuit down with a shampoo. There are a variety of specialty wetsuit shampoos, but as you know, I like to save money when possible. I find baby shampoo works just fine, and is substantially cheaper. Throw in a little baking soda as well to really neutralize any odors (read: pee smell).
You have two options for storage. Ideally, get a wetsuit hanger or a thick plastic hanger for storing your wetsuit in a closet. Make sure the wetsuit hangs in a form-fitting fashion, to prevent awkward creases. Don't use a metal hanger.
If hanging the wetsuit is not an option, you can fold it for storage. Be careful that you don't put any creases in the wetsuit, and minimize the amount of folding needed. For instance, try to fold it once longways (putting the arms and legs on top of each other), and then one more fold across the waist area.
Try not to squish it under anything. Permanent creases deteriorate the insulation quality of wetsuits.
It's a good idea to check for little holes or tears, while they are easier to fix. Most small tears can be repaired with neoprene cement. Loose patches and seams may have to be repaired with bonding cement.
If the hole is too large to bond back together, a neoprene patch is probably required.
Follow these basic guidelines so your wetsuit keeps you warm for years to come. And remember, an old wetsuit doesn't have to go in the garbage. Use them for activities where a wetsuit is needed but either the water is really warm (you don't need perfect insulation) or you don't want to use your new wetsuit. Pool dives are a great example.