The skytrain is composed of two lines interconnected at Siam Station: the Sukhumvit Line, which runs from Khu Khot to Kheha (Samut Prakan) and Silom Line that runs from National Stadium to Bang Wa. The BTS is connected to the MRT Blue line at Mo Chit (Chatuchak Park station on the MRT), Ha Yaek Lat Phrao (Phahon Yothin on MRT), Asok (Sukhumvit stationon the MRT) and Bang Wa (Bang Wa on MRT) stations and connected to MRT Yellow Line at Samrong station. Skytrain is also connected to the Airport Link that goes to Suvarnabhumi Airport at Phaya Thai station and to the Chao Phraya Express boats that you can take from Saphan Taksin station and Khlong Saen Saep boats that you can take near Ratchathewi station. If you change network, you will need to buy a new ticket. There is no common ticket yet.
Bangkok subway is called MRT. The metro line only covers a small part of the city even if connected to the BTS it offers more opportunities. The MRT is very clean, has air conditioned cars and is especially convenient to get to Grand Palace, Wat Pho or Chinatown.
The Chao Phraya Express boats run regularly on the river running through Bangkok. They go to all monuments and interesting places on the river shores including Wat Phra Kaew, Wat Pho and Wat Arun, the three must see temples in Bangkok. Fare is cheap.
Taking the bus to get around is clearly not the easiest mean of transportation to use for a foreign tourist in Bangkok neither the fastest one but it's the cheapest one. Bangkok bus network is wide and you can go almost anywhere but just need to be patient.