Stillwater has enough vacant properties that the rental cycle isn't impacted as strongly by the University as it might be in some towns with less apartment availability. As long as you look sometime during the summer months, you should have no problems finding a good apartment.
Stillwater landlords are generally pretty easy to deal with, and you shouldn't experience too many problems. Because rent is so affordable here, they're going to spend as much time assessing your character as they do assessing your ability to make the rent. Be polite, professional, and courteous in all your dealings with a prospective landlord.
Stillwater has what most towns would consider an amazing public transit system. Everyone affiliated with OSU can ride the bus for free, and other citizens of the town only pay 50 cents to ride the bus that serves the entire area. It even goes out to recreational areas. Many people here also drive, but things are close enough together that you can fare well on foot also.
Like in much of this part of the country, the support for college sports in Stillwater is very intense. Watching an OSU game is a great way to see high-level college athletics, as the Cowboys and Cowgirls compete in the Big 12. If you're looking to hangout with all the college students, Eskimo Joe's is the place to go. If you enjoy a slightly calmer Irish pub, head to Finnegan's Fighting Goat.