Some languages have a uniform system for expressing these differences, for example Mandarin Chinese is transliterated using a system called hanyu-pinyin (Chinese writing Pinyin system). Thai does not benefit from such a system.
So, in the "everything you know is wrong" category, everything I've read about Thai for the last 10 years, is not quite reality. Damn.
Then, the first thing about Thai is to learn the transliteration system used by people who do this for a living (as opposed to those who write books for people like me). I can't reproduce the characters here, so you'll have to use your visual imagination:
An "ai" sound (like in air) that looks a bit like e
An "er" sound (like mister) that looks like the number 2... but not quite. So we get words like "m22r". Your guess is as a good as mine.
An "aw" sound (like call) that looks like the right half of an "o"
And finally, a sound like "U" but said
with your mouth open as if smiling; represented with a u.
Try that "u" sound.
It'll bring a smile to your face. It has to, if you do it right.
It turns out there is a name for this transliteration system. It's called "Haas", and I even managed to find an inexpensive dictionary/phrase book that uses it.