I found the following three links very useful when figuring out wheel sizes and tire profiles:

Offset / ET Explanation
Offset Calculator for switching wheels.
Tire Size Calculator to see the diameters/circumferences for different tires.
Optimal and min/max wheel size calculator to choose wheel sizes for a given tire specification.

While bolt patterns specify which tires can bolt up to your wheel hub, there's a lot more to it - after the jump:

The variables of wheel offsets, widths, and diameters, as well as tire section widths, diameters, and profiles, inter-relate and dictate what you can (or can not) fit your car properly.

If the interior clearance (rarely a listed specification) is enough to clear your brakes, you still want a wheel that doesn't rub against your control arms or your shocks/struts/springs, and a tire that doesn't rub the inner and outer fenders over bumps and in turns. You generally want it to not stick out past the fender, might you might not care. You generally want it to not rub at full lock turning, but you might not care.

Apart from getting the correct bolt-pattern (obviously) I recommend confirming that a specific wheel clears your specific brake kit - either directly from the company manufacturing them (or selling them), or if you can see it mounted on another car.

For the highest performance when it comes to mechanical grip, you will generally want to pick the widest tire that can be mounted. As you can see from the calculators above, there will be a range of wheels that fit the tire; there will be some differences in appearance if it's thinner or wider. People will generally talk about tires and/or wheels when discussing the widest that fit. You'll probably end up playing back and forth a little with wheel and tire widths, given that not all wheels are available in the right widths and offsets (not to mention diameters), until you get something you like.

Of course, the simple alternative is to see if a lot of people doing the same kind of driving you do (track, autocross, street, etc) all seem to be running the same wheels without issue, and if so, just do that.