Generally, around September the northern parts of Thailand get less rain than the southern parts. So looking at the places you listed, Chiang Mai, Bangkok, Udon Thani etc will be less wet than, say, Phuket where the weather can often prevent speedboat services that take you to nearby islands / beaches. Koh Samui, Koh Phagnan etc are better off than Phuket because the latter is exposed to more rain clouds from the Andaman Sea (western side), apart from being further down south where the rain season lasts longer.
Rains over in Thailand aren't constant though - more like a heavy tropical downpours / thunderstorms that last about an hour or two and then stop. Days may not be as sunny, but you won't have to stay indoors all day. You may find flooded streets every now and then but nothing that will seriously impede your stay.