You may have already done your research, but:
Get a Budapest Travelcard which come is 24hour, 72hour and 7 day flavors. Do not get a “Budapestcard” unless you are going to spend all of your time in museums. They don’t otherwise pay for themselves and they mark you as a tourist. The Travelcard is good for all public transportation in the city limits.
Ride the historic M1 Metro but try and avoid the others. Not that they are bad, but you don’t want to spend your vacation underground.
Ride the trams as they are so much better than any HOHO Bus and cheaper too.
See Vaci utca, then don’t go back.
See the Buda Hill for half to a full day, then don’t go back.
Stroll Andrassy ut and the Pest River Front Promenade in the evening.
Hear some music. Try and get tickets to the Opera, the Liszt Academy, the Palace of Arts, the Budapest Operett, the Budapest Jazz Club or the Opus Jazz Club. These are venues where you will be in the company of Hungarians. The Church concerts can be good too but they are primarily for tourists. But, again, they are good. Budapest is about the music so this is a must. Get tickets early as they sell out months in advance. Splurge and get really good seats and dress accordingly. You can take your fur coat!!
Falk Miska utca for antiques
House of Terror is an excellent museum despite a kitsch name.
Take a picnic lunch to Margaret Island
Poke your head in the ruin pubs of the Jewish District (early in the evening if your are older, later in the evening if you are younger)
Hang out in a wine bar (Kadarka’s)
Eat, Eat, Eat, Eat…………………….
Oh, and if you are traveling in the summer, Budapest can be warm. Be sure your apartment has air-conditioning.
If you stay in Pest there is an excellent laundry service behind the Opera House. Beats the heck out of the tiny washers and 3 hour drying times inherent in most apartment washer/dryers.