I use smoked provolone and extra sharp cheddar that I get from the deli window. I love the combo of flavors, and they melt together really nicely. I always use a crusty sturdy Italian bread - none of that spongy soft cheap-o sandwich bread that goes flat like a pancake. There is no need to smash the sandwich down at all, just take your time over a medium low heat and let it slowly get toasted while the residual heat melts the cheese. I usually use room temperature butter, but I also sometimes use mayo, which works surprisingly well too. Actually if my butter is still too cold, I will use the mayo. You don't really taste it, it just spreads much easier, and seems to brown more evenly.
I play around with different cheeses from time to time, such as Butter Cheese, or Brick Cheese - both of which are very good. Around the holidays, I always wind up with a bunch of jars of that Harry & David pepper and onion relish. It's fantastic in a grilled cheese sandwich paired with extra sharp cheddar. Pesto or Olive Tapenade is fantastic with creamier tasting cheeses. Even fruit preserves, like Apricot, can be interesting when paired with cheeses.