How to use foreign words in English writing

If I italicize in the beginning like this: "Nine years ago, nonna (Italian grandmother), ..." then I don't have to continue italicizing the word

The approach here is not universal. Different style guides might advise different approaches.

Since nonna means grandmother, do I use the word the same way I would use grandmother? For example, would i say "my nonna" as opposed to "nonna" even though I call her nonna?

It's a judgement call -- both are options. Saying things like "my nonna" would give the writing a very intimate, personal tone. Using "my grandmother" would match a personal tone or a more detached tone.

English common nouns usually need a determiner like the, my, this. And in this case, if you don't capitalize nonna, you have made it a common noun and it needs my before it (even as a foreign word). English proper nouns don't need one.

So the first time you introduce her in the story, you should say

my nonna (Italian grandmother).

After that, you can call her Nonna, capitalized and without my.