THOU is the subject (Thou art…)
THEE is the object (I look at thee)
THY is for words beginning in a consonant (Thy hair)
THINE is for words beginning in a vowel (Thine eyes)this has been a psa
Also worth noting that “thy” is not an old-timey way of saying “the” either. Just wanted to add this because it’s the most common thing I see when people are trying to sound archaic.
ALSO.
THOU/THEE/THY/THINE is only for people you are familiar with. You wouldn’t use them to address the Queen, but you might use them to address your beloved.
YE/YOU is formal and official, and used to address someone of higher social standing.
YE OLDE is not a thing. It came from a time when the letters
þand y were indistinguishable when printed in period blackletter. So you’d still be visiting ÞE OLDE TAVERNE.