I live in Bangkok; I can mention some favorite places here.
My favorite stop is Jip Eu in Chinatown. They're not set up like a Western tea shop, so getting a feel for what they sell and getting into tasting in the right range can be tricky, especially related to language issues (they can speak it, but the amount of divide that remains doesn't help). That really goes more generally too, related to other shops, especially in Chinatown. At any rate they specialize in Wuyi Yancha, and have family ties to Anxi as well, and keep other different teas around. Per their input and other experience a lot of Wuyi Yancha are heavily roasted here; if you like that it works out well, and if you don't you would need to be clear on that early on to work through it.
The main mall shop people recommend is Tea Dee Zhang, out in the Thanya Park mall, way outside of the center of Bangkok on Srinakarin road. (
https://www.facebook.com/Teadezhang/) I've only been there once and they've moved locations since. They press their own sheng pu'er, and have for years, so they're into it on that level. Local mass transit doesn't go out there but taxis are inexpensive in Bangkok.
The main cafe is Double Dogs, on Yaowarat, the main street in Chinatown. To me it's not exceptional related to the teas they offer to drink there, although they are pretty decent. I haven't bought that much loose tea from them to ground an opinion about that. Beyond that, for other shops in Chinatown, K. Muikee Tea seems worth a look, and Sen Xing Fa carries a lot of pu'er cakes and shou mei, and the white tea cakes might really be worth a look. All of those places but Double Dogs will let you taste teas before you buy them. That wouldn't mesh so well with the theme of a cafe selling tea to drink samples they don't charge for first there.
Oddly I don't have much to add about visiting places in the North. Monsoon is a cafe in Chiang Mai that talks through a lot about "wild" forest grown teas. I've tried a white version and it was ok, interesting and pleasant, just probably a bit over-priced for what it was, per my take. Chiang Mai has been adding tea shops for a few years so there are a number of others.
They don't have a shop but looking up the Tea Side vendor would be a lead on the level of local teas that just don't turn up. Even related to that vendor it's still possible to dig deeper, there's still one more local tea layer to go. Check out this vendor for a sales option and more on what I mean, about what truly local style teas are like:
https://www.facebook.com/nanbangcha/
The large Chiang Rai oolong producers are ok for what that tea is. 101 is one of them, and there are only a half dozen or so. Per my experience Vietnamese copies of Taiwanese style oolongs can be better, and of course the higher levels of Taiwanese teas better yet.
It might be better to ask in a Thai Facebook tea group if you had a specific interest beyond any of that. I'll mention my favorite and the largest one here:
https://www.facebook.com/groups/Thailan ... sociation/
https://www.facebook.com/groups/ThailandTeaLover/
Don Mei's brother, Han, runs a shop in town, Seven Suns. It seems odd even mentioning it since I don't have the best impression of Mei Leaf as a source but Han is nice enough, and they do blends and sell matcha, so if in that area it might be fine to check out. If you want to buy the most expensive pot of tea you've ever had Peace Oriental has you covered; theirs start from $20 and keep on going. Oddly there isn't much beyond wandering around Chinatown to talk about. A few other places do come to mind but it seems as well not to mention them since that might just muddy the waters.
I've reviewed a lot of Thai teas; descriptions that tie back to a lot of what I've mentioned can be found in my blog, or discussion about Chinatown. The market alley on Yaowarat Soi 6 there is really interesting, with a number of tea shops on it, just none that warrant mention as a great tea source. All the same a short walk through that alley is well worth it for the visual experience and the smells.
http://teaintheancientworld.blogspot.com/