Hi, Abi, and welcome to the forum!
Where to go? That's a biiiiig question - Thailand has such a lot to offer!
In any event, I'd say that Bangkok is a must-visit - whether it's for temples, food, shopping, nightlife... and as you're likely to be passing through at some stage, it's no great hardship to get there. If you want some ideas as to where to go, we've got four whole pages full of ideas from Viator, who specialise in tours and things to do while travelling.
(We get commission if you buy using our Viator links, but it doesn't cost you anything extra - and even if you don't want to buy anything, a browse round will give you a good idea of the sort of thing you might want to do. Incidentally, if you click on any of the Bangkok tour links you'll be taken onto the Viator site, where you can browse for other locations in Thailand and elsewhere.)
British Expat: Bangkok tours with Viator
Can't really say about Hua Hin, as we've never been there; but people we know that have been there seem to have enjoyed it. It's a less in-your-face seaside resort than Pattaya, Phuket or Koh Samui (and it's also where HM The King spends a lot of his time outside Bangkok).
I'm not sure why you were advised against Chiang Mai! The North of the country is where it's least likely to get either hot or sticky - it's quite nice to get up into (or at least near) the hills.
If you want any more ideas about what to do around Thailand, we've got plenty of articles in our Thailand section on the main site
(which includes the Bangkok tours pages I've mentioned above).
As for getting down to KL and Singapore, the obvious way is to fly, whether by one of the established airlines or by one of the newer budget lines such as Air Asia. Rail is slow and either cheap and crowded or very luxurious and VERY expensive! (There's an article about the Bangkok-KL-Singapore Eastern & Oriental Express on the main site, again in the Thailand pages.)
Best of luck with your planning, anyway - and if there's anything specific you'd like to know about, please feel free to ask here and we'll do what we can to help.