Don't skimp on new tires... a tire that is worn will decrease performance and fun levels, dramatically. A worn knobby won't get you up a sandy hill or will let you get stuck in that mud that will take all of your energy to pull yourself out of. There's a huge selection, so do your homework. Read the reviews online and ask around. A local store should be able to guide you.
The basic selection of knobbies is between hard and soft rubber. The hard rubber is for hard dirt, rocks, hard tracks... while the softer rubber is more suitable for soft conditions: sandy soil, enduros.
Some of the softer tires are not road-worthy and marked as such.
I would prefer an intermediate cheap tire that can handle many conditions, like the Kenda Trackmaster. For general trail riding and some XC racing, I got used to the Michelin S12 XC - a soft intermediate.
The next thing about tires is pressure. It is important. Usually between 10-16 psi depending on the type of prevailing terrain. Lower pressure grabs well on loose surfaces but if you jump a lot on rocks, you may pinch the tube and get a flat, so you'd want more pressure. I usually just run 14 psi everywhere :).
I have heard of as low as 8 PSI on very muddy terrain and as high as 30 on road, for the same knobby tires.
On road you want more than 20, so... carry a pump and a gauge if riding a long distance to off-road? On the other hand, if you're into trials, the tire pressure can be very low and you shouldn't ride them on the street.
Balancing a tire is relevant for high-speed riding. Keep in mind that offroad tires are naturally un-balanced because of the rim lock.
Other dirt bike setup topics: