Basic BC Maintenance
Inflate the BC, check the dump hose for cracks and wear - replace as necessary (not usually). I would upgrade the assembly to a 1" diameter hose if possible, many of the smaller OEM dump assemblies are smaller (like 3/4") and don't dump air very quickly.
Leave it inflated over night. Loose air? You've got a problem, re-inflate and check in tub with suds for leaks. Some can be patched, some can't (bad seam, etc.), check with your dive store.
Replace the inflator valve on a regular basis. Just do it. They are the number one thing to go wrong. They aren't worth the money to service and can result in a catastrophic failure, with you making a rapid ascent if it sticks. Buy a good one (not the hard shiny plastic kind) for $30 and slap it on, takes a couple of minutes. There is a small pin that slides out to release the shoulder dump. You can use a small screwdriver or awl to slide it out. It does have one end that's thicker than the other, so take note. Be sure to double wire tie the new inflator on (double tie the dump hose onto the shoulder valve too).
Unscrew the valves, note assembly of washers, etc. Rinse inside with BC cleaner on a regular basis to kill bacteria. Check valve flaps for wear, replace as necessary (not usually). Squirt some silicone spray on the springs. Be sure they are reassembled correctly and are tight! While you're at it, spay some silicone on the low pressure inflator hose ends from the reg to your BC and drysuit. That is important to be done every few dives.
Check the tank bands for excessive wear and particularly the cams for cracks. If they are bad, upgrade to the metal cam kind, made by Scubapro, DiveRite etc.
Recheck by leaving air in it over night, and check again on a regular basis.
Always carry a spare inflator valve and some some ties. If you have a problem with it you can put one on in 5 mins while standing in your gear. I have saved many people's dives that way. I would also carry a spare tank cam band, if possible
Always check both the dump valves and inflator prior to diving. I have done 2-3 rescues of people whose valves were unscrewed, even slightly, particularly on the shoulder dump. Seaquest valves are notorious for it. (Note that I have a Seaquest BC and they are a good product, just watch the valves!
There, it took 30 minutes and now you know your gear!
Note: the above are general guidelines, your equipment may vary in what it needs. If in doubt, check with your local dealer.
See: ScubaDiving.com - BC care: What to do after every dive