There are several different sports you can get into with basenjis, with varying degrees of work and effort on your part.
Lure coursing is a lot of fun, and is pretty much an instinctual thing with basenjis. If your basenji has high prey drive (rips your arm off to dart off after squirrels while you're walking, etc), s/he will probably enjoy lure coursing. Since its instinctual, there's not a whole lot of training you can do, or need to do.
Getting started in lure coursing… go the akc website & do an events search for Lure coursing tests & trials in your state. It helps if you're willing to travel to surrounding states, as there aren't a ton of great places for coursing, so you could also search nearby states at the same time to find events. Also, go to asfa.org and click on the events link there. AKC & ASFA are 2 different lure coursing bodies, so the points you accumulate in one are not applicable to the other. By looking at events in your area, you can find names of the clubs that host events in your area. From there, google them for their website & contact info. You'll be able to find people in your area who can help you further.
Just doing an event search on ASFA & then googleing the host clubs, I actually found the Basenji Club of Southwestern Wisconsin http://www.bcosw.org/ and the Midwest Coursing Club http://midwestcoursing.homestead.com/. Those should be a good start for coursing.
If you want to get into agility, google agility training facilities in your area. Agility takes a lot of hard work and training for both you and your dog. You won't be ready to compete for a while, sometimes people train for a couple of years before they compete. However, its a GREAT learning and bonding experience for both you and the dog. Its a lot of fun!
There's also straight racing and oval track racing. I don't know much about those, but google NOTRA & LGRA & you'll find websites for both. Again, look for events in your area & then contact the host clubs.
There are a ton of events, and you can learn about most of them at akc.org. There's scent tracking, conformation, weight pulling, etc etc. You could also look into getting a Canine Good Citizen & visiting nursing homes or children's hospitals.
I'm sure somebody who lives in Illinois will chime in with what they do locally.