Well, you won't get a lot of current sport bike for $3k. You might be able to touch an early to mid 90's bike for that, but not many of those are in the $3k range. One bike that should be about that range but is a terrific bike is the Suzuki RF900. Kind of a sport tourer with about 80% emphasis on the sport part. Good looking bike too.
If you want something newer, an SV650 would be worth your trouble, but a few hundred dollars more. IF it were me, I'd just save until you can afford what you want. That's what I did this time around-it's not like you haven't ridden before, pick what you want in a bike and get it. If it takes a little longer, so be it.
Dave
BTW: Look in your area for salvage auctions. Those places can be a real gold mine for lightly damaged bikes that just need cosmetics to new bikes that were stolen and just need a few parts replaced.