We always use "åt mig" for "for me"? Or is there an instance that we used "för mig" instead?
Yes and no. Prepositions are a bit tricky to translate, since their usage may vary between languages. The preposition åt implies something along the lines of for the benefit/service of, i.e. that you do it for someone..
Depends on what you mean! :D
"Gör det för mig" is like saying "Do it for my sake" or "Do it for me" as in showing that new awesome dance move you just learned.
"Gör det åt mig" means "Do it for me [so I don't have to]".
Does that make it a little clear?
... and you could also use "till mig" or "med mig", but that would mean something nsfw?!
Sorry, but can't help wondering.
Yes, you could use those two too. Till mig = to me, as in I am the receiver of something physical. Gör det med mig is entirely contextual, so it's not necessarily NSFW.
Are there any resources in the discussions or on the internet going over those pesky prepositions?
Yes - it's the thing I notice the most often with non-native English speakers (who still have brilliant English) - they will get prepositions wrong because it's really just a random pick that is very hard to pick up unless you have been speaking that language nearly constantly for at least a year or two. For example "On Thursday, in April, at the park." Non-native speakers will often say "At Thursday, in April, on the park." or something.
Helpful hint for åt, för, and till (the three prepositions which translate to "for" in English) ...
åt is used if it's being done as a favor (as in this sentence) för is used if it's done before an audience (han sjunger for oss ... he sings for us) till is used if it's being done as a gift (han köper ett bord till henne ... he buys a table for her)
Would it also be "åt mig" if it was carrying it "to me" as in "bringing it to me"?
I had to use "bära" the other night, but I was wondering if I pronounced it correctly because to me, in most cases it seems to sound the same as "bara" (as in, 'just/only') but it's ä not a. Can a native explain if there is a difference or is it all context?
I think the first a in bara is more like "ah," while the ä in bära seems to often be more like the a in baa, although not so broad. At either times (according to the Duo pronunciation feature) ä sounds to me almost like "eh." I think, as in English, it depends whether or not the syllable containing the vowel is stressed or not, or in a word that contains stress, but I'm not sure. Sometimes (although not always) it's helpful to type similar sounding words into Google translate and then listen to them being consecutively pronounced.
ä normally = eh (ett äpple = ett epp-le), but it doesn't seem to be there are exceptions. e.g. "Jag är" = sounds basically like "Yarg ahhr" and not "Yarg air". bära and bara seem to be basically the same, though I haven't heard for a while.
Also, you'll notice I asked for a native speaker to explain it to me... Du svarade men det inte samma sak. ;-)
No, that would mean to me, but åt mig means for me in this case, like in 'so I don't have to'.
You can use it to mean 'wear' too, but that isn't used as often and it can be more formal. But Prinsessan bär en rosa aftonklänning 'The princess is wearing a pink evening dress' is certainly something you could see in the news coverage of some royal event.
Added that, but note that case in many cases can refer to different types of bags/things, like en portfölj 'a briefcase' or en verktygslåda 'a toolkit' or ett fordral (the case for e.g. a cell phone). So as a general translation, it's obviously a less good choice. The farther you go from the most common words, the bigger the risk that we haven't yet added a word as an accepted answer, even if we want to include it. We add words all the time so keep reporting (using the Report a problem button) but remember that whenever you use a rarer word, chances are your answer won't be accepted.
While I'm at it, the first translation for 'case' as an object would be låda, for instance en låda vin is 'a case of wine'.