Bangkok's Suvrnabhumi Airport, if it's the best known, is also the newest one. Opened in 2006, it supplanted Don Muang Airport on most international flights to Thailand and hosts domestic flights operated by Thai Airways, Thai Smile, Bangkok Airways and VietJetAir. Connected to central Bangkok by a dedicated subway line called Airport Rail Link very convenient to avoid traffic jams, several bus services to downtown or to some provinces (see below) and obviously taxis, it offers travelers all these options to get there or leave but let me detail them.
Detailed information, map, prices and schedules available on the Airport Rail Link de Bangkok
Count 1h30 roughly to get to Don Muang Airport.
Information and timetables on the page about the free shuttle between Suvarnabhumi and Don Muang
Convenient, he service has a very fluctuating travel time as it depends on the traffic. The journey can last from 50 minutes to more than 1h30.
Detailed information, timetables and fares on the page about bus S1 Suvarnabhumi - Khao San
Detailed information, timetables and fares on the page about page des Airport Limo Bus
Note that if GRAB, service equivalent to UBER, is available in Bangkok, do not order a VTC to pick you up at the airport, they are not wellcme there.
More information here on Bangkok taxis
If once you have landed at Suvarnabhumi airport, you want to go directly to Isaan (North-Eastern Thailand) or to some destinations further in the South, there are several buses available or private transport with drivers, including minivans if you are traveling in a group or family. The taxi is also an option.
Available routes are (click on it to see schedules and tariffs and book your tickets):
The most popular routes are (click on them if you want to book through my partner):