There are three major train stations in Budapest. Déli pályaudvar which is the biggest train station on the Buda site, Nyugati pályaudvar which is the biggest train station and Keleti pályaudvar which is the most frequented train station in Budapest.
For international trains I would start looking at Keleti pályaudvar as there are the most international trains leaving and arriving if you are going to the west. For trains leaving and arriving from the east the Nyugati pályaudvar is the right to look for.
Most international trains arrive and leave at Keleti train station. The trains coming from western countries therefore have to go through the whole city to reach Keleti railway station. The railway station can be reached with Metro M2. For a better orientation even though it is unfailable we give you a picture of Keleti pályaudvar.
The Nyugati train station is the station for domestic trains. Most trains here are going to the outskirts and smaller cities of Eastern Hungary. The Nyugati train station can be reached with Metro M3. You cannot miss the Nyugati pályaudvar but nevertheless we offer you a picture:
The Déli train station is covering the domestic western part of Hungary as well as some Internetional trains. The station is very rotten and ugly. Déli pályudvar can be reached with Metro M2. Deli pályaudvar is the biggest train station on the Buda side and every taxi driver knows but for a better orientation we give you a picture od Déli pályaudvar.