You could say "I'm going to go home" or "I'm going home", but not "I'm going to home".
However, grammatically you could say "I'm going to my home", and "I'm going to go to my home", but not "I'm going my home" (although you would instead change the word to house e.g. "I am going to my house").
"to home" no existe en general. Es una excepción rara que va sin preposición.
I would not say "Luigi's home". I'd say "luigi's house".
I'd just like to add for our Spanish speakers that two other, common options could be,
"We stopped by Luigi's / went to Luigi's last night."