English syllable structure is relatively liberal, but there are still a number of restrictions. Here are some of them. (Incidentally I suggest this is phonology rather than phonetics.)
With one possible exception, there can only be two segments in the onset (the bit before the vowel), and if there are two, the first must be a stop or fricative and the second is restricted to /r/ or /l/ (the latter not after a coronal) or a glide (/y/ or /w/). Thus in Greek words beginning with ps- or ks- ("x"), the first element is not pronounced in English, as they would not be acceptable combinations. If there is a coda (a consonant after the vowel, closing the syllable) then that is also subject to restrictions if there is another onset following.
Exceptions: /s/ at the beginning of a word before two consonants, as in "string". But only "s" can do this in English. Also you can have a pile of consonants at the end of a word ("twelfths"), but the extra consonants must be coronal.
Another important restriction involves stress. When for example a French word is brought into English, it starts being pronounced with English stress, and the unstressed vowels are accordingly reduced (e.g. "cabbage" or colloquial pronunciations of "garage"). Notice also in this last example that the final consonant becomes an affricate, as the corresponding fricative does not exist in English at the end of a word (only in the middle of a word like "measure").
There is more to be said, but this is the kind of thing to look out for.