I just returned from a 3-week trip to SE Asia, ironically. We were in Asia from Nov 17 through Dec 8.
We started in Singapore (3 days), then went on to Bangkok (3 days), Krabi/Railay (7 days), Chiang Mai (4 days), and finished in Hong Kong (4 days). I posted a few photos from the trip in
this thread if you're interested.
I recommend finishing the trip in Hong Kong, personally... it worked very well for us, because there's several direct flights from HKG to YVR. Singapore, obviously, is much further (another 5 hours or so beyond HK) and therefore a more arduous trip... I prefer to have the worst flights/travel to begin the trip, when there's something to look forward to when you finally step off the plane.
Since you've been to Singapore already, I won't bother telling you about that. (Spoiler: it's awesome.)
Bangkok is pretty chaotic, busy, and difficult to get around. Many people told us we didn't need more than a few days there, and I agree... it's just a huge city with terrible traffic, which makes getting around the city a pain at times. Don't get me wrong, there's stuff to see and do, but I'd recommend just a couple of days in Bangkok at most.
In the Fall, when you're going, it's the dry season on the Andaman coast, which is where we went... we were on Railay Beach (flew into Krabi) and it was *awesome*. Top-notch beaches (the country's best, I was told), beautiful resorts, lots of sunshine and things to do.... it's beautiful there, as you'll see if you check out the other thread with my photos. You can fly into Krabi or Phuket from Bangkok for <$100, and from there it's fairly easy to get to beaches, resorts, the Phi Phi islands, etc. If you like beaches, it's a must-do.
The opposite coast (the Gulf of Thailand side) experiences the "wet season" during Fall, and is drier in the Spring... I'm told it's also amazing (Koh Samui looks great), but being on the beach in a torrential downpour isn't that ideal.
Chiang Mai is like a greener, calmer, more beautiful version of Bangkok. It's got all the cultural tourism stuff you could want -- hundreds of wats and templates and history -- but without all the scammers, traffic jams, etc. Unlike Bangkok, there are no huge office towers or modern mega-malls... it's still a big city with great nightlife and all the conveniences you're used to (including a great airport), but it's more authentic and pleasant than Bangkok. Stay in or near the Old Town and you can walk to most of the city's top tourist spots. Outside Chiang Mai, you can visit one of 30+ elephant parks, hike in the jungle, take a river boat tour, ride an ox cart, see a monkey show, etc etc.
Hong Kong is quite simply a top-5 "must visit" city... in my books, it's up there with Paris, London and NYC. Incredibly unique city that truly has an "East meets West" vibe. It's got the jaw-dropping skyline and city sights, plus the history, incredible shopping, and tourist spots of all sorts to keep you busy for as long as you're there. We stayed on Nathan Road (Kowloon), and it was a great location... but anywhere in the central areas of the city (both on HK island and Kowloon) you're usually within walking distance of everything, or you take a cab (or train) and it should be quite cheap.
If you've got enough time (and you should def give HK at least 3-4 days), I strongly recommend a visit to Stanley and some of the lesser-known areas of HK -- it was so much more beautiful than I had imagined (reminded me a lot of the Monaco or the Riveria). We also did a day trip to Macau (there's a 50 min ferry leaving every 15 mins), which was pretty cool... not a "must do" in my eyes, but it's definitely a bizarre place that's worth seeing if you've got time.
Everything in Thailand is cheap, especially if you're willing to take the "budget" option on various things (ie. eat at street vendors, stay in smaller cheaper hotels). We prefer a more upscale travel experience, so we didn't go for any super cheap hotels/tours, but our beautiful 5-star hotels in Bangkok and Chiang Mai were both less than $130/night including tax and breakfast. Outside of big hotels/resorts, a good meal shouldn't cost more than $7-$10 per person, including beer & tip.
Hong Kong, on the other hand, is really expensive... probably 10-20% more than Singapore (based on my limited experience). Especially food. Our hotel was very nice (but not insane) and it was around $300 CDN per night... expect to pay at least $150/night for a decent hotel in a decent location.
Despite it being the "cool" season, it was hot as shit in all the places I've mentioned above, except for Hong Kong (where it was around 15 degrees & cloudy/windy)... I only wore pants/sweater in HK. Everywhere else, I was in a shorts and sandals, and usually rocking a sweat-drenched shirt. Chiang Mai was milder thanks to lower humidity, but daytime highs were still 25-30*C.
If you're a less experienced traveler, or you want an 'easy' travel experience, Thailand is for you. It's way more developed than I had imagined, and I always felt safe, even walking around unlit streets at night. English is everywhere, things run more or less on time, and we never got ripped off or scammed... if you're not completely retarded, you can travel around Thailand without any problems at all.
I've never been anywhere else in SE Asia, but we talked with many people on our trip who had been elsewhere... Vietnam/Cambodia are probably the next most popular spots, with relatively developed tourism industries in both. Laos and Burma are your go-to spots if you want to get off the beaten path and enjoy some adventure/real cultural immersion... both are pretty undeveloped, so if you want to see them before they become as cliche as Thailand, now's your chance.
In three weeks, though, I'd personally suggest: Singapore, HK, and one or possibly two other SE Asian countries. Getting around from place to place takes time -- budget a full day of travel if you're getting on an airplane, even if the flight is only an hour long. Our 3-week trip was perfect, and would've been rushed if we'd tried to cram any more into it... you could spend less time on the beach (we had 6 nights in Railay) and visit more places with that time, but I wouldn't stretch it too far or you'll be exhausted and ready for another holiday by the time you're finished.
Have fun.