A very easy and enjoyable day trip is by HEV train (a local rickety little train line) to Szentendre, about 30 minutes or so from Budapest and situated on the Danube. Others recommend taking a boat ride for the return but I didn't do that. You take the metro to Batthyani Ter station and then buy a ticket to Szentendre--be sure to ask for a return one as well, if not taking the boat back. Very cheap.
Rick's Budapest book covers it in great detail and I thoroughly enjoyed spending about 5 hours there on a warm summer afternoon. It's a quick walk into the little town, I was there on a day when there was a street fair going on with all kinds of wonderful hand-made crafts (it's an artist's colony) and delicious food, and there are several restaurants with outdoor seating. Little twisty lanes lined with shops and churches, very pretty and peaceful. I found some excellent souvenirs there, and something for me in an antique shop. You can tour the churches and just wander the charming little streets, and the highlight for me was visiting the little Margit Kovacs museum, she was the first female Hungarian artist to receive wide acclaim for her whimsical paintings and pottery figures.
There are other towns to visit, of course, and I loved Budapest so much that I'm planning a return trip next spring and will be taking the train--so far in my tentative planning--to Pecs, Gyor and Tata as I recently discovered my ancestors came from there and there's several sites of interest in the town as well.