Caveat: this is just my own experience learning Thai, I may have gotten some stuff incorrect and I am not a Thai nor language teacher for sure. My intention is not to teach Thai, but to supplement whatever you are currently already reading with my own experience.
The Thai tonal system is the first thing anyone will notice when hearing a Thai person speak or when you start learning thai. For a Chinese or tonal speaking language person, the tonal system is not difficult to learn. One of the unfortunate thing about being a Singaporean is that, I have a good command of English and at the same time am confortable with speaking Chinese, but most English language books are written for English-speaking-only audience. Worst still a lot of equivalent tonal system or phonetics using English alphabets are used (just like the over liberal use of romanji in a lot of Japanese language books). It really screws up properly language learning.
The Chinese tonal system have 4 tones (and 1 more short tone): mid, rising, falling and low
The Thai tonal system have 5 tones (in a different order): low, mid, high, falling, or rising
The 4 Chinese tones matches the 4 similar tones in Thai, so that should not be an issue.
The high tone is actually similar to a rising tone, except that it starts at a higher tone compared to rising which start from your normal tone. In other words, you can hear a very distinct rising of tone when speaking a rising tone, whereas a high tone have a less distinct rising tone. So if you string a bunch of high tone words together, you will sound like a shrieking banshee!
Its interesting if you actually ask your Thai friend difference between high and rising, they may not be able explain to you the difference between the two, they just know how to pronounce it and can tell that you pronounce it wrong, but not the mechanics of it.
In conjunction with tonal system, understanding live/dead syllable and long/short vowels also helps you pronounce the tones more accurately.
More specially, rising and falling tones (except if tonal marks are used) only occurs for live syllable and dead syllable with long vowel (but NOT vice versa). Short vowels with dead syllable are almost certain high or low tones. Although I am not an expert, but I suspect even with the tonal marks, this rule will retain (with exceptions of course).
The reason for the above is simply a case of economy of speech. Rising and falling tones need time to rise and fall during the pronunciation and only live syllables or long vowels allows that to happen. Again, this does not work in reverse, that is you cannot have a live syllable with high tone, the tone drags too long and risk sounding like a rising tone.