When you arrive in Bangkok at Suvarnabhumi International Airport, if you need to a connecting domestic flight, you can catch your flight there or you might have to travel to the Don Mueang International Airport Terminal.
Domestic Airlines include
Upon arrival, if you have a connecting flight, make sure you understand exactly which airport, Suvarnabhumi (where you landed) or Don Mueang Airport your connecting domestic flight is leaving from.
If your flight is leaving from Don Mueang, you will need to arrange transportation as their airport is ~1 hour+ away and more time depending on how heavy the traffic is.