If you can, I highly recommend going to Tahoe or Virginia City. Both are about an hour drive from Reno, but they are great cities to spend the day in.
Downtown Reno is a lot of fun to walk around too. There are some really great restaurants in the casinos. Wingfield Park is a beautiful park to go to. It's near Dreamers and other cafes and a movie theater and shopping. You can go tubing or kayaking on the River, or just wade in it.
If you are up for shopping, the Summit Sierra is a good place to go. It's an outdoor mall with a few restaurants.
You can also go to the Planetarium (at the University).
A lot of the casinos usually have some kind of performer so you might want to check to see who will be in Reno during your stay. The Grand Sierra Resort also has a $3 movie theater. It also has a bowling alley and an arcade.
If you are into golf, there are several beautiful golf courses in Reno.
Dreamers is a great place, as is Waldens and the Chocolate Bar. The Chocolate Bar has delicious desserts but I wouldn't have any of their entrees. If you like sandwiches, I recommend Jimmy John's. There are some really great restaurants in the casinos too. The Silver Legacy and the Atlantis are my favorite. I've heard that the restaurants at the Peppermill are good too but I haven't tried them yet.
I'm trying to think if there are any events going on in early June. Art Town doesn't happen until July and I don't think Street Vibrations will be in June either. Wild Waters might be opening in the beginning of July.
I'll get back to you once I come up with more.