Here is the most green, economical and actually fun way to see a lot in 20 min for the price of a fridge magnet. Hopping on tram 2 is considered one of the most beautiful and cheapest ways of sight seeing in Budapest, as it rides by the riverside of Danube, driving you past the most historical buildings from Jászai Mari Square to the National Theatre.
It’s free if you own a daily, weekend or any kind of public transport pass. I highly recommend to look to your left and right too, as the tram goes by the Parliament, the Gresham Palace, Chain Bridge and even under Elizabeth Bridge. On the other side of the Danube the Buda Hills can be seen as well as the Castle District and Gellért Bath.
If I have to go during the day to this area, that’s the first thing which comes to my mind. Even if it is longer I still take this tram. Tram 2 is not the only public transportation that gives an alternative way of exploring the city, check the BKK Boat Service which offers new and other possibilities too.
I recommend you take a return journey because both sides are crowded with nice spots!