I'm going to jump ahead and guess it's not a Floyd Rose style of trem, or your concerns would be different than they are. :p

I posted this link in another forum. This is a pretty impressive vid on tuning a vintage (Strat) style trem so it stays in tune.

http://www.youtube.com/watch?v=O0Oyc6slYRc

Again, not knowing what guitar you have, you can try some lubricant on the nut (shave graphite off a pencil, or buy a specialty lube), make sure you have good tuning machines. Locking Sperzels are popular.

Put enough springs and tension so that the bridge holds tight to the body when at rest. You won't be able to pull up on the trem, but the tuning will be a lot more stable.