We were in Budapest for a week and spent many evenings walking the bank of the Danube and criss crossing over the 4 main bridges. The city is built up around the Danube and (nearly) all the main attractions can be seen from it.
Tram no 2 goes up the Pest side of the river and is a cheaper alternative to an official tour as you will see many of the sights and the ticket price was approx. £1.20
We also took a 10pm boat cruise on our first night - saw all the major attractions and got a brief history about each attraction while listening to headset. The weather was great as it was after sunset so it was cool enough to sit and enjoy the cruise. More