In this case, yes, da can mean to. If you've got any familiarity with French, it is somewhat similar to chez.
Alla nonna would literally be to grandma, but da can be used to mean to somebody's house, to somebody's place, as in the following examples:
- Mangiamo dalla mamma (We're eating at mom's place, mom's house)
- Andiamo da Mario (We're going to Mario's)
- Loro vanno dai fratelli (They're going to their brothers' place)
Hope this helps :)
- Not using a definite article with a singular family member is only with the possessive: mia nonna; le mie nonne; tuo fratello; i tuoi fratelli.
- Just as French as "chez so-and-so" to mean "so-and-so's house", Italian has "da+definite article so-and-so" to mean "so-and-so's house". This was explained on this page a few years ago.
The translation solution suggested by DL for this particular one is so restricted that it is counter-productive, my Italian friend tells me that this can be an acceptable translation for so many ways of saying that one is going to 'see grandma', and that it's not one of those where you can translate 'word for word' as it were.