Where to go in Thailand in September?

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.

September is the rainiest month of the year. See rainy season info and a precipitation map.

The north and northeast get somewhat less rain than the south. If you like beaches, Koh Samet (an island in Rayong province, about 4h east of Bangkok) probably gets less rain than southern islands, and is no less beautiful.

Rainy season should not be an impediment to travel, if you're flexible and patient. Downpours are usually strong but brief. The rain seems difficult to predict -- overcast sky does not indicate it will rain anytime soon and that you should stay indoors, and a sunny morning is no guarantee it won't rain 2h later.

I was in Bangkok this morning. It raining very pleasantly for about an hour the previous evening. Of course, several days earlier, I got caught in a downpour that was like standing in the shower; that also lasted about an hour.

My point: unless you are going to be doing something that involves walking on dirt (e.g. hiking), which this time of year would involve wading through mud, the rain is something between a minor interruption and a pleasant break in the day.