Meet me in Montauk.
Male - 21 Years Old
First off, I've never been here during the summer (this year will be my first one) so this is all second hand knowledge. But campus itself is usually pretty dead during the summer besides a few summer classes, professors staying for research, and construction. That being said, State Street/Capital Square area is pretty much your go-to place. Just down the street from campus and on an isthmus between two lakes, lots of retail, museums/art gallery's, a few venues, lots of bars. Memorial Union and Monona Terraces are supposed to be great places to grab a pitcher during during the day if that's your thing. So if you're near campus all that should be pretty accessible.
I think your plan to kind of scope out the place next month or whenever is a good idea. Depending on when you're here you might get a really good glimpse of Madison at its best, right at the beginning of summer. I've spent all of one week in the NY/NJ area so I don't really know what it's like, but Madison is a passable small- to mid-sized city, especially if you're in your twenties.
Edit: That's a little bit off campus, a few miles, but in the right direction as far as nice residential goes. Sorry to say I can't give specifics for that area as far as things to do go. On the plus side, Madison is a great city for biking if that's your thing.