Go over to the Buda side to Castle Hill. This is a non-negotiable if you are in Budapest. Fisherman's Bastion offers beautiful views of the Pest side of the city and the stunning Hungarian Parliament Building. It also connects all of the castles on the hilltop and is a scenic and enjoyable walking tour of Buda.
When you are walking back to the Pest side stop at Platan Grill and order the goulash. This little spot is easy to miss so keep your eyes peeled.
Once you make your way over to the Pest side of the city find a Ruin bar. (Most of them are located in the old Jewish Quarter.) What began as an underground club experience is now one of the top attractions when in Budapest. There are no neon lights outside these clubs so don't be afraid to ask around. The original ruin bar is Szimpla Kert and one of the most popular!