Budapest is a great "tourist" city—
classic sights to be visited, plenty of accommodation and restaurants, lots of English spoken and of course a river running through the middle—
or rather between the cities of Buda and Pest.
With architecture on the grand scale lining both sides of the Danube, walking the banks or better still cruising the river is the way to see it. A lot of the multi-day cruises on enormously long river-boats start or finish at Budapest, and there are many 1–2 hour tours being promoted the minute you set foot in the country. But the best bargain in Budapest is the public ferry system, operated as part of the city's transit system, the BKK.
Although you can buy individual tickets, the best deal is to get a 24 (or 48 or 72) hour BKK Travelcard that gives unlimited access to the whole transit system—
the metro, busses, trams, some other trains as well as the ferries.
We bought our Travelcards at the Deák Ferenc tér metro staion where a very helpful agent photocopied a ferry schedule for us. We then took the metro to Batthyány tér, where we left the station, walked the few yards to the BKK quay and got on the next ferry heading upstream. This took us past the Parliament building and Margit sziget (Margaret Island) and eventually to the end of the service at Rómaifürdő. We then returned, heading downstream, and eventually disembarking at the Petőfi tér quay.
You'll not get the token free-drink, on-board toilets or tour commentary in umpteen languages that come with the commercial tour boats. But you will get to see the same great views of Budapest, and get to sit inside or out, and be able to get on and off freely at any of the stops.
One thing that nearly caught us out… If the upstream and downstream ferries arrive at the same quay together they will both dock, one beside the other, both pointing upstream, and you may need to climb through one boat to reach the other. It's not obvious what is happening and we very nearly didn't board, thinking our boat had not arrived yet.