I actually stayed with my girlfriend who lives outside of Venice about 1 hour by train, so in the 3 months that I stayed with her, I got to go to venice about 5 times. I know this is gonna sound cliche, but try not to go during the summer. You've seen how many tourists there are in the summer, well this is a tiny group of islands that is known as one of the #1 places to visit in Italy, so it's just wall to wall Americans, Brits and Germans.
I think 3 weeks should do it, but to see more of Europe you'll need more time. Even though I haven't gotten to go there yet, I hear that Cinqe Terra is amazing and beautiful.
I was considering visiting NZ, but from what I've heard there's mostly alot of drinking, smoking weed and hot girls, which would be fine except that I might get myself in a bit of trouble without my girlfriend there.