Just read your intro post.. Looking forward to reading about your experiences..
If you plan on staying for just 3 months, you can get a "tourist visa" from the Thai embassy in your city (New York was it?)..
The tourist visa should be good for about 60 days. At some point within that 60 day period, you'll need to get an extension. You'll be able to do so at the immigration office from within Thailand. What area are you going to be in? And have you decided on which gym yet?
In Phuket for example, the immigration office is located in "Phuket Town" and you'll be able to get a 30 day extension for a 1900 baht fee.
You mentioned that you won a lot of money playing poker so it sounds like you have the freedom to do and go wherever you want. There's a pretty good chance that after spending a few months here, you'll get comfortable and you won't want to leave.
So, to be on the safe side, you might also want to look into getting yourself a "type B non-immigrant visa" (or an education visa depending on which one you're local embassy is willing to give). I'm in Thailand now with the type-b (which is actually a work visa)..
This will allow you to stay in Thailand for a maximum of 15 months (if you time it right). Every 3 months however, you will need to leave Thailand a re-enter. You can be gone for 5 minutes, it doesn't matter. On each re-entry, you'll get a 90 day extension stamp on your passport.
In order to get the type B non-immigrant visa (or an education visa), you'll need to get an invitation letter (and perhaps the school registration papers) from the gym that you'll be training at.
Let me know if you have any other questions..