The Millenium Underground was the first underground on the European Continent, opened in 1896. It is now part of the UNESCO World Heritage. The innovation that made it unique all over the world was that the underground railway used electricity instead of steam motion. If you use this line you can easily visit the most beautiful sights of the Pest side.
Budapest has an extensive transit network which features subways, trams, and buses. Public transportation is inexpensive and is quite easy to navigate. Since the city is relatively small, many attractions are within reasonable walking distance of The Deák Ferenc square (the very center of the city), but of course if you’re getting tired of constant walking, I give you now a couple options to reach the most beautiful tourist attraction the easiest and cheapest way.