Before I was tonsured a Reader in the Orthodox Church I had to learn the eight Byzantine tones. The easiest way was to just memorize the eight Resurrectional Troparia. Once you get these down you can chant just about any text. It’s a matter of doing it over and over again.
As for Gregorian–it’s easy to learn square note notation. I have a Psalter with settings of all the Psalms and that helped. Now I can plainchant just about anything.
A good free resource for chanting of all types is Unmercenary Sacred Music. He uses regular western round notes, so as a musician this may be easier for you.
My prayers are with you.