Will it be correct if I use 'I'll go to check it out' instead 'I'll go check it out'?
What about I will go and check it out?
Is that also correct?
It's fine.
Context is always important; labelling is rarely important.
Why is it better? I try it out with other verbs and it doesn't seem better to me:
I'll go [to] find out why.
I'll go [to] drive the car.
as opposed to
I'll go to chaperone the kids.
does sound better... In this latter sense, “I’ll go to drive the car.” sounds like I’m attending because no one else knows how to drive.
It doesn't seem better to you because, being American, you're probably used to the "go do something" form.
I think "go to do" or "go and do" are more consistant with our use of other verbs in general, for example "Come and do the washing with me." I'm not sure if American English uses "Come do the washing" or "Come watch TV with me". If it does, it's introducing a new grammatial structure: verb of movement + main verb, which is not traditional in English.
"Go do that; Come do this; Sit eat your dinner; Stay do your homework ..."
So, the structure "Go do something" is an exception to traditional grammar, and perhaps this is why it is judged by some not to be as good as "Go and/to do something".
It certainly does.
I don't make the leap from using "come" and 'go" to the other verbs. It's more like "Initiate this action" either toward me (Come watch TV with me) or away from me (Go scoop the cat box). I don't know anyone who is 100% comfortable with that use of come/go would be even slightly comfortable with "sit eat" or "stay do."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.