How to pronounce 儿 in conjunction with other characters

This is indeed a very complex question and not easily answered fully (although what other people have said will still get you very far in practical terms). San Duanmu (2007) offers a fairly good introduction in his The Phonology of Standard Chinese. It goes far beyond what most casual learners want to know, but it should be interesting for anyone who wants to delve deeper into this question (and many others, the books is an introduction to Chinese phonology in general).

Wikipedia has a very useful page on Erhua. I don't know how to type IPA here and I'm not sure whether the OP knows IPA, so I'll summarize the rules informally here, writing the approximate erhua pronunciation inside quotations marks. (Note that this does not represent the pinyin spelling of erhua words; those rules are different.):

• Final i and n are deleted: 把儿 = ba + er -> "bar"; 伴儿 = ban + er -> "bar"; 盖儿 = gai + er -> "gar"
• Main vowels i and ü are retained, and the "er" sound is added at the end of the syllable: 气儿 = qi + er -> "qier"; 劲儿 = jin + er -> "jier" (the n is deleted by the above principle); 裙儿 = qun + er -> "qüer", 驴儿 = lü + er -> "lüer"
• Certain vowel sounds merge. These rules are complicated; I refer you to Wikipedia for more details.
• Final ng is deleted, but the vowel becomes nasal. I've left this rule until last because (as a native English speaker) I find these sounds difficult to pronounce. Wikipedia, or the sound files mentioned in other answers, will give you a better sense of how these vowels sound.

I haven't seen the rules formalized, but it seems to me if a word ends in a vowel, simply add the "儿" on the end just like 花儿. If the word ends in the letter "n", drop the "n" and replace it with the "儿" sound, like 馆儿.

I can't access the sound files that trideceth12 linked in his answer, but I've heard 空儿 pronounced almost like 2 separate phonemes, rather than like one smooth sound like 花儿 or 馆儿. In this kind of case, where the word doesn't end in a vowel or "n", it seems like you just add the 儿 sound on the end without changing the root word at all.

Audio clips of ”有空儿“ being pronounced can be found on this page. From my (limited) listening experience the 儿化 "儿", as with regional accents in most languages, is not entirely predictable.