That's going to mean different things to different people. All things being equal and if you have no particular driving interest then I suggest you stay within the zone defined here: https://goo.gl/maps/zeG69ZcZSfD2 Its mostly "Budapestian" (is that a world?) with some tourist overtones. At one end you will see Deak Ferenc ter. If you were to calculate the center, based on walking time (or walking and metro time), of all the major points of interest in Budapest it would be close to Deak Ferenc ter.
The significant transportation lines that you will be interested are the M1 metro (the oldest on the continent of Europe), the 47/49 Tram which runs from Deak Ferenc ter to the Great Market Hall and the River. and the 2 Tram which runs the Pest side of the Danube; and to a lesser degree the 4/6 Tram. Find those on a map and you get a pretty good idea why the area works so well. The zone I point out is along the Andrassy ut corridor which is a beautiful walk, but the M1 metro runs right under the street if you just want to make time. The M1 will connect you with all the above ground transportation you will want to use. Both the M1 and the Trams are fun and not merely transportation.
The area bounded by Deak Ferenc ter on the east, the river on the west, and the Erzsébet híd (bridge) on the south and the Chain Bridge on the north sort of defines the Tourist Hell of Budapest.
Okay, Tourist Hell is strong. This is the zone that exists for the sole purpose of serving tourists. Tourist restaurants, all the big name brand stores that you find in every tourist center, hundreds of souvenir shops and everything substantially more expensive than Budapest as a whole. It, in and of its self is a touris attraction, but nothing in the zone is of particular “Hungarian” or “Budapest” interest other than the river front which is quite beautiful.
The apartment you should be able to find within the zone I described for about 60 euro a night. Also significant to the area I suggest is that in the evening you can walk among the Hungarians, eat in Hungarian restaurants and drink in Hungarian wine bars; all at Hungarian prices. But I am terribly biased……..
utca = street
ut = boulevard or major road
ter = square
And this is just my personal preference and where we stay when in town 3 times a year. There are a lot of good places, just dependent on your preferences and taste.
For me, mid September to mid October is the absolute best time in Budapest.
For a cultural experience visit the Operett, Opus Jazz, Ruin Pubs, a working Synagogue, Bath House, Wine Bar, drink Palinka, hear a Klezmer concert, and explore District VIII.