I've been to Thailand many times, have family there, and have traveled domestically there over much of the country. It's not a small place, and it can take time (more than you might guess) to get around.
This is not the answer you want to hear, but I've got to be honest: personally, I would not recommend going to Thailand on such a short trip (10-12 days total trip time). Keep in mind that it can easily take 2 days to get to Thailand (depends a lot on where you are starting from and your flights), and another 2 days to get back. The international date line plays some tricks (you appear to get back home on the same day you depart Bangkok, sometimes within just a couple hours of your departure..) but no matter how you slice it, you're flying to (roughly) the opposite side of the planet, then flying back. That eats up some time. You're starting with "10-12 days" and you have 3 destinations...do the math. Ouch.
From Chicago, I would try hard to get a one-stop flight, with a connection in Tokyo (or possibly Taipei); it's still a long, long way to go, but you would need to find every way possible to reduce your travel time (ORD-TYO-BKK is not a cheap flight, but it's an efficient routing and Japanese airlines are lovely). Avoid additional stops en route. I hope you've done some long flights before (after this trip, the short hop to Europe will seem like nothing at all).
So my suggestion would be to either (preferably!) find some more days for your trip somehow; reduce the scope of it (flying around domestically will burn most of a day each time you go, that's another day used each way); or reconsider your plan. With your plan - at least 3 locations that are far from each other (Bangkok, Chiangmai, and some island) you are running short on usable time to be in any of them. Get out a calendar that you can write on (in pencil) and write out your days. It will be sobering.
Good/bad months...their "summer" (hottest weather) is our (North American) Spring. I would avoid March/April/May unless you really enjoy intense heat and humidity. This is of course relative, to me it's always hot in Thailand, but in April it's OMG hot. Depends on your tolerance for intense heat. Thai New Year (Songkran) is also in April and it's a very popular time for domestic tourism, so prices spike and availability disappears, another reason to avoid that month. Rainy season traditionally is around September, but recently may be extending from August through October (thanks to climate change). It's not raining all day long but when it does rain, it really comes down hard and there's often localized flooding.
Samui is OK but it's fairly touristy and not really the untouched, "pristine" tropical getaway that many people dream of. You can certainly snorkel there (as an avid snorkeler, I'd say that there's better snorkeling much closer to home, eg Mexico). That said, Samui is easy (and quick) to get to, and popular for reasons which are obvious. Personally, I prefer the southern islands on the Andaman Sea (Indian Ocean) side - they're not quite as developed as Samui (with the exception of Phuket, which I would avoid - it's Thailand's Cancun). The islands south of Phuket are lovely, many with jaw-dropping scenery and are much closer to the stuff of your fantasies. Takes more time getting to/from them though, and that's something you will be struggling with.
Hope some of that helps.