The skytrain is composed of two lines interconnected at Siam Station: the Sukhumvit Line, which runs from Wat Phra Si Mahathat to Kheha (Samut Prakan) and Silom Line that runs from National Stadium to Bang Wa. The BTS is connected to the MRT, Bangkok subway, at Mo Chit (Chatuchak Park station on the MRT), Asok (Sukhumvit stationon the MRT) and Bang Wa stations. If you change network, you will need to buy a new ticket. There is no common ticket yet, the two metros are separate entities. 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.
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 Hua Lamphong station and to Grand Palace, Wat Pho or Chinatown since the end of July 2019.
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.
Only Thai city for the moment to have a metro (there is metro project in Phuket), Bangkok is the city in Thailand that offers more choices of public or individual transport (metro, bus, boats, tuktuks...). No, there are not just tuk-tuks and pink taxis to get around Bangkok !