Well, Jake, I guess you can't "help" me further because I can't imagine anyone being confused about what "quickly" is modifying in "I'm going to the store to quickly buy some milk." The idea is that you're going to be in the store briefly because you're just shopping for milk, not the week's groceries. It obviously has nothing to do with how quickly you get to the actual store. Now, if you sincerely don't see that, then you can always insert the words "in order" in front of "to" to make it even clearer: "I'm going to the store in order to quickly buy some milk" - but that's understood.
As krs was getting at, once the "to" part of the infinitive is introduced, it logically links the following adverb to the infinitive. Like I said before, whether to split the infinitive is only a question of style. The basic meaning of the sentence is not affected.
krs said
In the example you just posted:
I'm going to the store to quickly buy some milk.
I read that as two actions:
"I'm going to the store" and (at the store) "I will quickly buy some milk"
You then replied
Ah, but that is NOT what I meant. I meant that I was running to the store as fast as I could, would then buy some milk. Won't rush to check out, I like to browse. I might even buy more things! But I want to get there quickly as I am concerned that the store might close before I get there.
So your interpretation was incorrect because of the ambiguity of what I wrote.
Respectfully, I'd assert his interpretation is correct based on the structure of the sentence, and if that's not the meaning you intended to convey, then you did not formulate the sentence properly. You should have said, "I'm quickly going to the store to buy some milk" or "I'm going to the store quickly to buy some milk."
This is no more ambiguous than "The man bit the dog." No English speaker is going to say, "Wait a minute. Did the man bite the dog or did the dog bite the man?" The word order in English makes the meaning clear, even though you wouldn't expect a man to bite a dog. Same thing with a split infinitive. The pattern of "to" + adverb + infinitive is abundantly clear - they are a unit.