For $2K you should be able to find a 600CC sportbike from the mid to late 90s in half decent shape.
Kawasaki Ninja, Suzuki GSXR, Honda CBR, Yamaha YZF
You would do well to try and save another $1k.
With $3k cash you should be able to bargain your way onto something between 2000 to 2004 in good shape.
Just rememebr your after purchase expenses. You will need a helmet and jacket at a minimum.
Then you'll need to change the oil & filter, air filter, coolant flush, maybe brakes & tires, spark plugs, and battery.
Getting an older bike may mean you would spend another $1k in repairs and maintenance anyway.