While there are a lot of variables, and sometimes it'll take some trial and error, there are some general rules that can help get you off to a good start. Here are some general guidelines, along with a few myths or fallacies to avoid:
1. The pressure printed on the tire is the MAXIMUM, not a recommendation. Some vehicles, often heavier trucks, will run maximum tire pressure when heavily loaded. But for most cars running the maximum is too much.
2. The right pressure depends on the tire size and the vehicle. There is no such thing as a specific recommended pressure for a certain brand and model of tire.
3. The manufacturer's recommended pressure on the door jamb is a great place to start. It will list a recommended pressure or pressures, and also usually tell you the load index of the original tire. If the load index of your new tire is lower, you may need to add some additional pressure; if the load index of the new tire is higher, you can still run the recommended pressure.
4. Keep an eye on treadwear. If the center is wearing faster than the edges, drop the pressure. If the edges are wearing faster than the center, raise the pressure.
5. If you have plus-sized your wheels, it is often a good idea to add some extra air pressure to help protect them from potholes.