First off, even if you're not a cast iron fan - don't get a non-stick grill pan. They're not meant for high heat. The cast iron version is highly recommended.
The trick to using a grill pan on the stove is to realize how it relates to cooking on a real grill. Treat it as you would if you were grilling over direct coals with the lid open on a grill. If you would finish the meat strictly like that on the grill, then you could finish it fully on the stovetop. Think burgers, sausages, stuff like that.
If you would move it off to indirect heat on the grill to finish, then that's when you need to consider it moving it to the oven. Sear a chicken breast a couple of minutes per side and then put it in the oven to finish.
As far as what food, I can't imagine anything that you would normally cook on the grill that you couldn't fashion to fit on the grill pan. Obviously, food will lose that smokey flavor. Consider smoked paprika or liquid smoke to add a bit of that flavor back in your food.