Aw man, this brings back all sorts of memories for me. I lived in Chiang Mai for about 8 months in 2012-2013, and I miss it a lot.
It seems like you already have a good handle on Thailand, and I second a lot of the advice already given. Here are a few of my thoughts:
--Chiang Mai is great. Sure, it's overrun by tourists, but there's so much to do and see in the area and the vibe is so good if you actually get into it, that it's worth it. Really fantastic trekking, lots of animal-related stuff (this may be controversial, but we visited the tiger sanctuary there and thought it was fine...we did a lot of research first and deduced that the "Tiger Temple" somewhere closer to Bangkok was home to a lot of animal abuse, but the one near Chiang Mai was decent...it's possible that I'm wrong and contributed to an unethical practice, but the animals seemed content and playful and well cared-for. And also goddamn adorable.), outdoorsy things, great food. All of the street food is excellent, and there are a ton of good restaurants. If it's still open, go eat at the Cat House restaurant and say hi to Jacky, though I think she was getting married to a Canadian and moving there. Avoid the "Zoe" complex of bars, they're terrible. They do have fire spinners some nights though, I actually took lessons from a chick that worked there. Check out the "sticky waterfall," that's a fun spot to go with friends and have a picnic with locals.
--Best spot off the beaten path in Chiang Mai is an old red rock quarry full of amazing blue-green water. I can't remember the name, but at least when we were there it was almost only locals that knew about it. South of the city a ways. Cool and refreshing water, great cliffs to jump off of, not many people, another good place to bring food for a picnic. We had a scooter, you could rent one if you're brave enough to handle the highway or try to arrange something with a songthiew or maybe the owners of your hostel. Wish I could remember the name, it might be hard for you to find but way worth it. Google around, it's probably better known by now.
--If you need a scooter or private transport, look up Jeff at Red Ride motorcycles. He's an American expat that has the nicest bikes and very reasonable rates for private cars. We rented from him our entire time there, and did border runs to Burma with him for not much more money than it would take on a bus, in half the time. Very reasonable and dependable guy.
--I'm not sure if it's already happened this year, but look into the Yi Peng festivities (where they do the huge sky lantern release). There's a small, expensive one for tourists, and a huge, free one for locals (go figure). We did the latter and it was truly awe-inspiring. Loi Krathong, during the same holiday, is also fun (people release little floating floral arrangements with coins on them into the river as a form of prayer...little kids swim in the river and steal all the coins).
--In Chiang Rai, check out the black house. There's the white temple, which is interesting and beautiful if extremely inauthentic, and then there's this other creepy-ass spot that everyone misses. It's not a temple, per se, but it has some similar and fascinating architecture, with a ton of wood carvings and really ornate stuff, as well as an endless supply of taxidermied animals. Weird, dark vibe. I think it's just some artist's private home/gallery, I can't remember. Definitely worth a stop.
--It doesn't get much more burnery than Pai, and there are lots of outdoor activities like rafting if you're there in the right season. There's a little music festival called Shambhala that happens there, run by Japanese hippies. It might be around the time you're going, though maybe more like January or February. We took one of the minibuses between Chiang Mai and Pai and didn't think it was that bad, but we have generally low standards for ground transportation.
--Koh Lanta was my favorite island by far. If I recall correctly, the ferry port is on the northern tip of the island, and there are more or less three stretches of beach down the west side with plentiful lodging. The middle one is the most laidback and burnery, with good vibes and cheap lodging and fire spinners on the beach. Lots of "special" shakes on the menu, but please be careful--I had one one day and it was awesome, and had a different one a different day and wound up spending a scary night in the local clinic. Enjoy, but be smart about it and look out for your partner. PM me if you want more info. My lady went scuba diving in Koh Lanta (I'm not certified

) and said it was really, really excellent. Also a good island for renting a scooter and just driving around, not a lot of traffic and beautiful views.
--You probably already know this, but avoid Koh Phangan like the plague. There are the full moon parties, and then different "moon" parties about three days out of the week, every week ("new moon" "half moon" "jungle moon" "shiva moon" "no moon" etc.). We went to one of the "other" parties and it was horrifying. We spent about an hour around sunrise cleaning up trash on the beach and got some other people to help, but there was just so much filth it was really saddening. I can't even imagine a full moon party, especially during the holidays, but we had a big group of friends do NYE there and had their room broken into and all of their valuables stolen. Just stay away.
We spent about a week in Burma, I'm jealous that you'll have three weeks there. We flew Bangkok to Mandalay (make sure you're up to date on the visa process, it's probably changed since we went but at that time it was easy to procure in Bangkok in about 24-48 hours, and you could only fly into Burma unless you were just crossing to a border village and coming back to renew your passport stamp). From there we went south by bus to Inle Lake, Bagan, and Yangon.
--Mandalay -- Genuine city, very friendly. Checked out the Mustache Brothers show, which was cute and an interesting piece of modern history. They've been in the news recently, google them if you haven't heard of them. There was a temple on a hill where young Buddhist monks come to practice their English with tourists, we really enjoyed that. Big market (one of our favorite things to do when traveling). Not a ton of attractions, but we were okay with that because people were just so nice.
--Inle Lake -- Take a boat tour, cheaper if you coordinate with other people from your hostel. You'll stop at markets, villages on stilts, unique fishing practices, some "factories" for textiles or hand-rolled cigars. There's supposed to be some really good trekking in the area, but we didn't have time to check it out. Most fun thing we did was rent bikes and ride out to a little Buddhist monastery, where there were little novice monk kids that wanted to play with us.
--Bagan -- Stunning, not to be missed. Definitely catch a sunrise there. I think we maybe rode around on bikes one day and on a donkey cart another day, I can't remember. It was hot as balls at all times though, that made it difficult to enjoy in the middle of the day.
--Yangon -- Decidedly meh, this was our least favorite place but we were also pretty burned out and ready to go home by the end of our time in SE Asia so we might have judged it unfairly. Schwedagon (or Schweddyballs, as I think of it) Pagoda is pretty awesome, but that's it. We sort of hid in our disgusting little overpriced hostel room. One other thing that we didn't care for much in Burma, as long as I'm complaining, was a lot of the food. We
love all manner of cuisine, but most of the curries and main dishes in Burma were unpalatable to us--too pickled and fishy and weird. The many different kinds of "salads" were good, just order a bunch of small plates of those rather than hoping for dank Thai-style curry.
Have fun, SE Asia is amazing.