It's worth walking over the Chain Bridge because of the view. It only takes a few minutes, because the Danube isn't all that wide. The approaches to the bridge are slightly busy, of course.

First you could visit Vörösmarty Square (Vörösmarty tér), at the end of the yellow line or the subway/underground (called kis földalatti in Hungarian). It's a nice place, and you could walk up and down Vörösmarty utca (street), a well-preserved pedestrian zone and shopping district.

It's also easy to get to Vigadó tér on foot from Vörösmarty Square. The view is lovely there, and you can get the No. 2 tram/streetcar to the Chain Bridge. It's also where you should catch a bout to take you on a two-hour cruise of the Danube up and down Budapest, preferably at seven p.m., when the bridges and buildings will be illuminated when you are on your way back.

After crossing the Chain Bridge, you would find yourself at Clark Adam tér. This is where you can catch a ski lift up to the Buda Castle (Budavár), more of a palace than a castle really, which contains the National Gallery. Once you are up there, walk all along Castle Hill (Várhegy), with it's lovely Medieval buildings. All the restaurants here are expensive. Another bridge worth walking on is the Margit (that's Margaret) Bridge, not because the bridge is lovely but because of the view from the middle of the bridge, where there is a sort of observation deck, and because that's where Margaret Island (Margit Sziget) is, with it's lovely park for walking and having picnics in.

Take the no. 4 or 6 tram/streetcar to the middle of the bridge. Parliament can be visited at Kossuth tér, easily accessible on the red underground-subway line. It's impressive. Heroes' Square (Hősök tere) is worth a look, also on the yellow line, as is the Opera. See a ballet at the opera; it's lovely inside. We just saw Swan Lake, which was just amazing.

Another lovely building is the Iparművészeti Múzeum (this is like the museum of crafts or something). Worth a stop, and not far from the 4 and 6 tram/streetcar lines.

Cruises can be taken to other great places nearby: Szentendre, Visegrad and Esztergom. I heartliy recommend taking just a circular route, setting out to the north in the evening, and swinging back in the dark. Seeing the banks of the Danube from a boat at night, with all the illuminated monuments and bridges drifting by and towering over your head, is well worth it.

Enjoy your trip to Budapest!