Let's say I've just been food shopping with my girlfriend and I want to say "give me the bag" so that I can carry it for her. Could I just use 「くれて」, or would I use something else?
Since you are trying to take over a heavy bag from your wife, I'd think more appropriate expressions are:
バッグ(bag): combined with a hand gesture, just saying "bag", maybe just
ん, or even not saying anything at all, communicates the right message. I prefer this the best, because the terseness shows that you and she are on the same wave length that doesn't nee too many words, which is generally seen as a sign of a long intimate relationship. otherwise known as 阿吽(あうん)の呼吸
持つよ([I will] carry [it])
頂戴(ちょうだい)(pass [it to me]): similar terse forms that just uses a verb and omits everything else. This works, too.
下さい(please give [it to me]): a slightly formal word like this, when combined with exaggerated intonation, can add playfulness.
I don't think くれて in a grammatically correct Japanese in this context, though there's くれてやって/くれてやんな, which is more blunt/aggressive form that translates to something like "just give it up already"
if you're shopping and you want to ask waiter to show something you want,
You can use
if you want to say give me the bag in japanese when you're shopping.
You can just say
in your sentence.
you also can take a look into other questions already asked and their answers in Stackexchange.
I understand that you are asking for a casual command form, but I have found the expression
持とうか / 持ちましょうか
to be the most useful because you can confirm that they want you to carry it by just saying this first, or you can assume they want you to by saying this while reaching for the bag.
In the context of carrying a bag for a girlfriend, I don't see why ください or any variation of it would be necessary if spoken with normal tone (i.e., excluding the usage Kawaguchiさん describes).
Well, I guess the most "normal" saying is that you say something like:
But what I usually say with my wife (日本人) is something along the lines of:
I know it's probably not the best and most accurate saying, since it also changes with the social relation of the speaker/listener, but this was always understandable for all 日本人 I talked to (her parents, only know 日本語, relatives, etc.)