Well, I am learning gaelic irish for a couple weeks now, and I can't just keep in mind for example lets say various Nations like Netherlands, ditnguishing between country name, nationality and languages. And then are just different ways to say things an never can seem to get the right grip
For example, what's the difference between "theacht tú abhaile" and "Tháinig tú abhaile" and which one is the corect one to use?

As far as I'm aware, theacht tú abháile is not correct. The past tense version of tar is Tháinig, so tháinig tú abháile would be used.

You can use teacht in a sentence like Tá tú ag teacht abháile (you are coming home)

or:
(Ba mhaith liom) tú a theacht abhaile. = (I'd like) you to come home.

True. Thanks for showing an example with lenition. I couldn't think of it. I'm going to blame it on the fact that I say, and often write, tíocht instead of teacht.