Sorry, @fitterman, almost forgot to get back to you! 
There are just a few, at least that I can think of right now. But note that none of these are mandatory, especially since we can simply just drop the first vowel. So, you may not always hear these “liaisons” even when they’re possible. Also: 1) I’m basing this mainly on the pronunciation patterns I hear around me and use myself, in Lisbon + surroundings; 2) check out the interactive table at the end of this article for sound references of each vowel type (closed, medium, open):
Unstressed ending E vowels followed by medium or open vowels E, A or O → These will be closed E vowels, which may sound like the English e when followed by the aforementioned vowels. I’d say this happens more with monosyllabic words like que, de or se, or enclitic pronouns like -te or -me, than with longer words.
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O que é que achas? (What do you think) → O qu[ee] é que achas? (modified first vowel) or O qu’é que achas? (omitted first vowel)
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Tu és de onde? (Where are you from) → Tu és d[ee] onde? (modified first vowel) or Tu és donde? (omitted first vowel, which forms a contraction in this case)
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Eu quero-te aqui (I want you here) → Eu quero-t[ee] aqui (modified first vowel) or Eu quero-t’aqui (omitted first vowel)
Unstressed ending A vowels followed by medium vowels A or O → These will be medium As, which tend to be omitted themselves and also change the following vowel into its open version. This may happen with any word in general, not just short ones:
– Medium A + medium A = merge into one open A
- É a avó! (It’s grandma) → É [à]vó! (open A)
– Medium A + medium O = merge into one open O
- Paga o que deves. (Pay what you owe) → Pag[ó] que deves. (open O)
Ending S consonants followed by any vowel sound → These tend to go from a sh sound to a zz sound, as is usually the case anytime the letter S is between vowels. The letter S is explained in detail in this Learning Note (The Letters S and C) and in this other forum topic (How to pronounce "s" at the end of a word).
- Os homens são altos. (The men are tall) → O[z] homens são altos (note that the letter H is always mute, so the vowel sound after it is all that matters)
We don’t, or at least we don’t have any layman’s term for it.