You can hop on an ICE directly at FRA airport to Freiburg (2hrs) for a lovely small town holiday and walking in the Black Forest, or you can go up to Cologne for big city museums, etc.

Although I like Frankfurt, I would probably suggest either of those if you have 3 days to spend. (And there are other easy options from FRA, e.g. Heidelberg, or a bit further off Basel.) If you do stick with Frankfurt, look for a hotel out in Sachsenhausen, on the other side of the river, it's a nicer area.