you are very correct not wanting to use google translate, most of the times it has funny results.
allow me tp rephrase you, so it can be more understandable.
1. Μαθαίνω ελληνικά για να μιλώ μαζί σου. mah-thEH-noh eh-llee-neekA yia nah mee-LOH mah-ZEE su. where capital letters is the stress, the ''th'' as is thunder, the ''yia'' as the Greek yiayia. it says ''i am learning Greek so i can speak with you''.
2. you can say: Θέλω να σε επισκεφθώ, thEH- loh nah seh eh-pee-skeh-FTHOH, it literally means ''i want to visit you''. in Greek the actual word for visit is somehow more formal, so since she is your grandmother you can say something along the lines ''i want to see you'' or ''i want to stay with you'', the-loh na seh d(th)o or th-EH-lo nah mEE-noh mah-ZEE su. the d(th) means that the ''d'' is the letter equivilant to the Greek ''δ'' but it is always pronounced as the ''th'' in words like then, there , thus and so on.
3. Ελπίζω να είσαι καλύτερα, eh-lpEE-zoh nah EE-seh kah-LEE- teh-rah, i hope that you are better.
you can also say Περαστικά, peh-rah-stee-kAH which i dont't think there is an one word English equivilant for that , it is as the english ''feel better'', we say that wish especially to people who are having health problems.
how sweet of you wanting to put a smile on her face. good luck!