The Airport Link is connected to the MRT subway station Makkasan (Phetchaburi MRT station) and BTS skytrain, at Phaya Thai station. It is also possible to take the Klong Saen Saep boat shuttle if you get off at Ramkhamhaeng station. Up to you to choose according to your final destination.It is also possible to directly catch a train to continue your journey at Lat Krabang and Hua Mak which are stations on the Railway Eastern line.
The trip from Suvarnabhumi airport to Phaya Thai lasts 30-40 minutes. If you are 4 people or more a taxi can cost less but you may got stuck in trafic jams.
The Bangkok Skytrain is called BTS. The trains have air condition and let you go over the heavy road traffic in the capital but you can not go everywhere even if connected to the Chao Phraya Express baots, it can let you go to major temples in Bangkok.
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 !