We used to live in Berlin, I'd recommend:
A boat tour on the Spree
For the Wall, go to the Bernauer Strasse memorial instead of Checkpoint Charlie
The zoo is nice, it's next to lots of shopping as well, and there's a trendy rooftop restaurant called Neni next to it where you can watch the animals below
Depending when you go in June, the outdoor pools will be open -- the one in Monbijouplatz is really nice, with a park next door for a picnic
Ice cream! Berlin has the best ice cream and it's really cheap, there are shops all over
I see someone mentioned Tropical Islands -- this would be a great day trip, go early! It's really something.
If you haven't booked a place to stay yet I'd recommend something in the Prenzlauerberg - Hackescher Markt corridor, it's one of the most charming parts of the city with loads of restaurants and shops