Usually the simplest words have the stupidest origins.
Get showed up in the thirteenth century, coming to us from the Old Norse geta, where it basically meant the same thing as what we use it for. It’s from the Proto Germanic getan and the Proto Indo European root ghend-, to seize or take. Interestingly enough, in Old English get was almost only used in parts of words—you know, like in forget. But the Old English word for get, gietan, is not where the Modern English get arrives (like I said, it’s from Old Norse). They do however happen to have the same root.
Now, I already mentioned forget, so let’s get into it. Unlike get, its origin is in Old English, where the word is forgietan, which is just forget. The for- is thought to mean a kind of negation, so it’s something like losing or being taken away or the opposite. Basically, the opposite of getting is forgetting.
And there’s one other word related to get: beget. Not that we ever use it much. It’s from the Old English begietan, and means something like acquire or obtain, but involving some effort, and it wasn’t until the thirteenth century that it had anything to do with reproduction. Misbegotten (which I think is probably used more these days than beget) is actually derived from beget. It showed up in the sixteenth century from misbeget, which was a word at one point even if it’s not now. And when it did first show up, it meant bastard, illegitimate (presumably referring to offspring), or unlawfully obtained (presumably not).
TL;DR: Forget isn’t from get, but they are both from the same word, so my opening comment was right.
Tony Jebson’s page on the Origins of Old English