"go look for" -- is it ungrammatical?

[Sweetpumpkin]

Hello, guys!

I found the following sentence in a dialogue:

We'll go look for an apartment to rent.

I know "go look for something" is acceptable in speaking situations, like "go see a doctor." However, I'm not sure whether it is grammatical or not. As far as I know, it is wrong because a bare infinitive cannot be followed right before another bare one. So, if it could be corrected, can I use like "We'll go to look for~" or "We'll go and look for~"? And if 'go' is possible in that way, 1) how about come? "You can come look for it.", etc. 2) what kind of verbs can follow right after "go"? Only see and look are possible? How about "go get it" "go take them" ?

I'm looking forward to your answers. Thank you in advance!
sweetpumpkin

[Barb_D]

See that cute boy over there? I dare you to go speak to him. Oh, go take a long walk off a short pier, would you? We'll be with you in a moment. Why don't you go have a seat over there for now and we'll call you when we're ready.

It has this sense of movement in the command.