The easiest and most comfortable way to discover the city is to take a sightseeing bus. There are several companies offering tours through the city with different sized buses, from large 40-50 people buses down to rather cars for 6 or 8 people. Also very popular in Budapest is the so called Hop On Hop Off Bus where you buy a ticket for 24h and can take the bus anytime you and simply hop on or off at any bus station.
During the bus tour you always have a guide either private talking to you directly or you get headphones telling you more about sights. The advantage of a private guide is of course that you can ask questions as well as that you always get the information in the right moment which is not always the case with the headphone solutions which are coming from a computer probably. Nevertheless it gives you good infotmation about the sights around you and an indication if something is interesting for you or not. So if you pass the Parliament in Budapest, which defintitely looks impressive, you might want to take a tour to get more detailed view on it.
To discover Budapest by Bus is probably the most relaxing way so if you want to have a comfortable afternoon this is the way to go. But of course you can choose also more active tours such as the Segway Tour or the Walking Tour.