The International Trains arriving from Western Europe are quite ok. Clean, fast, relatively cheap and a train is always romantic. Flying by airplane is fast but takes from you the opportunity to get used to the new environment step by step. When I travel for example to Prague I prefer to use train or car even if its longer then flying. To come here from Vienna is 45 mins by plane...but 2,5 hours by train. From Prague to fly is 1,5 hours, by train is 6-7 hours. And I still prefer to use train.
The railways stations are not the nicest places in Budapest. The Western and the Eastern Stations look great and are attractions as architectures but inside they are dirty and messy with old-fashioned and low quality services.

If you arrive to Budapest by train you might be approached immediately by taxi drivers and people offering accomodation. Just ignore them...
You also might be approached by people offering money exchange. Just ignore them...

Why? Because those taxi drivers are mostly private sharks charging you a lot, if its a real, legal hotel and hostel service they do not have to hunt for users like this...and the city is full of banks , ATMs, and official money exchange services so you dont need to exchange money on the street like this.

Its sad to say but arriving by train maybe take some photos of the Station building and then leave the place without using any service there.

The schedule of international trains to/from Budapest you can check on ELVIRA.

InterRail:


Budapest Eastern Railway Station

Keleti pályaudvar is also a station of the M2 (East-West) line of the Budapest Metro (subway) and from about 2009 on, it will be a station of the M4 line, too. So even if its not right in the downtown from the station you can reach your destination quite easily.

Budapest Eastern Railway Station (Budapest Keleti Pályaudvar) is the largest among the three main railway stations in Budapest. It can be found in the 8th district (Józsefváros), on Baross Square. Its not really the downtown but even not far from it. The 8th district is the poorest in Budapest so arriving to the Eastern Railway Station do not expect a fancy, clean area. Though the building itself is interesting and impressive.


















The Eastern Railway Station in 1905.

It was constructed in eclectic style between 1881 and 1884 and was one of the most modern railway stations of Europe in that time. It was planned by Gyula Rochlitz and János Feketeházy. The station's length is 93 m, the platform length is 80 m, underground depth is 14 m.

The main façade is adorned with two statues of James Watt and George Stephenson.
source: wikipedia

Some photos of the Eastern Railway Station. To see them in larger version click on the photos:




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Budapest Western Railway Station

The station was planned by August de Serres and was built by the Eiffel company. It was opened on October 28, 1877. Previously another station stood in its place, the end station of Hungary's first railway line, the Pest–Vác line (constructed in 1846). This building was pulled down in order to construct the Nagykörút.

This is the third larger Railway Station in Budapest located to the 1st district in Buda. It was build in the 70s and was considered as a modern Station that time but today it is in a critical condition. Recently its under reconstruction but the station is operating. Though only a few internation trains are arriving to here (mostly to Croatia and Slovenia) but the station might be important for you if you want to go to the lake Balaton for a few days. You can reach there by 1,5 hours so its not a big deal to make an excursion for fun.

The Southern Railway Station is the METRO terminal for the M2 line. So even if its not in the downtown it has a good link to the city.