I was passed this link by a German friend, as I think I'm known amongst friends for my opinions on Osnabrück - I too was an American living in Osnabrück for a summer, just a couple years ago. I don't really have a high opinion of the town, honestly I found it to be pretty boring.
While it's billed a student town, most of the students I knew seemed to live somewhere else, and the student culture in the town seemed non-existent. The town seemed overrun with old people, and the whole place seemed to shut down early.
That said it's not all bad - certainly first and foremost you should use it as an opportunity to travel elsewhere. As others have mentioned, Amsterdam is quite close, in fact when I travel there from where I live now, Bremen, you transfer through Osnabrück. Berlin is definitely doable as a weekend trip (though of course you'd just scratch the surface of the awesomeness that is Berlin on a weekend trip). Also, the train ride south to Munich is beautiful if you took a trip down there. There are lots of other interesting towns, nearly every German town of any size is an interesting visit. Lots of options for travel, I highly recommend taking advantage of them.
In Osnabrück itself there are also some gems that I found while there. There's Rampeldahl, a little Hausbrauerei that makes hearty German food, decent enough German beer, and a house made Kräuterlikör that comes garnished with a slice of sausage and mustard. There's also a chocolate shop, Leysieffer, which is a chain across Germany but this location is somewhat unique in that it also sells ice cream, and they are constantly making the cones throughout the day so they're fresh and the chocolate chili ice cream is actually spicy!
As for culture, there's the Felix Nussbaum house, which is a really stunning museum (the building itself as much as the contents). The Kunsthalle Osnabrück is worth a look, again it's interesting as much for where it's at (in a former church) as the art inside. The Rathaus is an important historical site, as it was where half of the Peace of Westphalia was signed, which is responsible for concepts of state sovereignty as we know them today.
For nightlife, Lagerhalle can be a nice place to grab a couple beers, they have a little more variety in drinks than your average bar. Hyde Park has been mentioned - this actually was a place I hadn't been to until after I moved away, but recently I saw Gogol Bordello there and quite liked the venue, other than how far away it is from the city center.
Other than that, my only other tip is not to live at the Apartmenthaus am Westerburg, they're assholes.
Beyond that, not sure what your interests are. I've lived now in several German cities and visited dozens more, so I'd be happy to pass along other suggestions more related to your interests to help fill your calendar for the time you are here in Germany. :)