For a complete experience, visit both parts of the city, both are divided by Chain Bridge with Buda(Castle Distric, more historical) on one side and Pest(city centre) on other. Metro and buses goes from Keleti train station to Széchenyi Chain Bridge and reaches it in 10-15 min, you can take taxi – 9 min or walk 40 min.

BUDA - > Visit Castle District, the district, called Várnegyed in Hungarian, is situated on a hill at the Buda end of Chain Bridge.  Getting to Castle District: The whole district, which is an UNESCO World Heritage Site, reflects medieval and baroque Budapest with many fine buildings and narrow cobbled-stoned streets with footpaths winding around the hill. The main historical attractions in Buda Castle District include: the Royal Palace, Matthias Church on Holy Trinity Square (Szentháromság tér), and Fishermen’s Bastion from where you can see a fine view of the Danube river with the bridges, the Parliament building on the Pest side, and the rolling Buda Hills.

PEST - > start at Széchenyi István Square (formerly Roosevelt Square) – Pest bridgehead of Chain Bridge (Lánchíd). Mainly shopping streets and lots of statues. You are gona be walking/drivng all through this part from train station in order to get to historic Buda part across the river.  

Or just relax in Gellért Thermal Bath, Budapest with hgeothermal pools, baths etc. Opened 6am – 8pm. Very architectural structure as well. Price with locker 20CAD. Adress: Kelenhegyi út 4, 1118 by Liberty Bridge (Szabadság híd) http://www.gellertfurdo.hu/