Budapest is a stunner of a city. Along with Warsaw, one of the best value places to visit in Europe.
Hotels, markets, restaurants, historic buildings all in very close proximity and cheap as chips. Also, as someone else mentioned, the local wildlife is very rewarding to study in as non-creepy a manner as you can manage. Casual appreciative glances being enough as another will be following along shortly.
If your mobile and don't mind a hill, stay close to the Castle district in Buda. If you're less mobile or a bit of a foodie, stay close to the Palace district in Pest, then visit Jozsefvaros for the food and drink.