I figure I better get around to the other half of Budapest before I get any further along. So, Pest: the flatter, more urban center. Or at least the part that would be more recognizable as a sizable city on its own. An isolated Buda would feel like a historical relic, largely, whereas Pest could function fairly independently.
I of course - like most of Budapest's non-wealthy, non-geriatric population - live in Pest, though far outside the fashionable areas. Those fashionable areas are concentrated relatively near the river, roughly coinciding with the distance between those bridges outlined in the Buda entry.
The hub of these is Deák Ferenc Tér, the main (and only until Metro Line 4 is completed at some unknown theoretical date in the future) transfer point for the Budapest Metro. It feels much different than many similarly emphasized squares do in other European cities; this likely stems from the fact that a significant portion of the area is taken up by a skate park and combination indoor/outdoor nightclub space. (As well as a fairly creepy shrine to Michael Jackson that has on at least one occasion had an impersonator moonwalking near.) Needless to say, it's a bit less formal.
From Deák, the various (fun) areas of Pest are all easily accessible. Andrássy út is the most exclusive of these, a wide, tree-lined boulevard in the manner of Las Ramblas or the Champs-Élysées. It prominently features old houses from the turn of the century, many of which now house embassies. Beneath it runs Metro line 1, the oldest in continental Europe, which I will surely ramble extensively about when I talk about Budapest's transportation network.
Andrássy ends in Heroes Square, which in turn leads to the city park, a large expanse with replica castles, baths, and a circus in addition to those more predictable park amenities.
To the west is the Hungarian Parliament, likely the most recognizable building in the city. On our last trip, we repeatedly joked that the Hungarian financial crisis (as of 2009, the current Eurozone crisis of course has much more wide-reaching origins) was likely caused by the building's upkeep.
Right near school is the major cathedral in the city, St. Stephen's. I could probably think of something more substantial to say about it, but I see it 5-10 times a day, so let's face it, I'm pretty jaded by now.