Merhaba
I will visit Turkey next month I was learning for  3 months now

I would like to know in Turkish daily life which for of these is used and are they have the same meaning.

Thanks

Sen çay istiyor musun?
sen çay içmek ister misin?

From grammar perspective,

... çay istiyor musun?

 -yor is (present) continuous form, so the question sounds like more for "now"; "do you want tea now?"

... çay ister misin?

-r is aorist form, and also has an "offer" meaning in kind form. So it sounds more like english offer: "Would you like Tea?"

with the word "içmek", it is added "to drink", so;

çay içmek istiyor musun?

sounds like more "do you want to drink tea now?"

çay içmek ister misin?

is more an offer; "would you like to drink tea?"

After these explanations;

when I read your questions, I imagine some stories.

Çay istiyor musun?

The story is; I am going to get tea, do you want tea too? so I am asking it in Turkish; "Çay istiyor musun?"

Çay içmek ister misin?

I am host, you are welcome to my house. I have tea, but not boiled (cooked) yet. So I offer you tea and I may boil it if you want. So I ask:

"Çay içmek ister misin?"

Does this explanation make sense?