Well discussed, but maybe a quick answer here will not be in vain.

'Auld lang syne' is literally 'old long since,' but that doesn't help unless you know that 'long since' means, more or less, 'the past.' You can't do word-for-word translations and capture meanings very easily, so here's commentary:

"Should auld acquaintance be forgot, and never brought to mind?
Should auld acquaintance be forgot, and auld lang syne?"

It's a rhetorical question:
"Do you think we should forget old friends and never remember them?
Do you think we should forget them, and the days gone by?"

Burns does the same sort of thing in "A Man's a Man" --

"Is there for honest poverty that hangs his head and a' that?
The coward slave, we pass him by -- we dare be poor for a' that."

"Is there anybody who, just because he's poor, hangs his head?
We walk past such a coward slave -- WE dare to be poor (we're poor and we're proud)"