If you do decide to do a road/trail bike, something like a Yamaha TTR250 is a great bike to start on. I had one as my first bike, road it for two years and did nearly 15000km on it during that time with no problems whatsoever.
I have since had a 2003 Yamaha WR250F and now have a 2007 Yamaha WR450 which I have motard wheels for. The maintenance side of owning a modern 4 stroke is a bit of a pain as I'm servicing it every 1000km, whereas the older 4 strokes, you can do service intervals at every 4000km.
A good 2000-ish TTR250 or equvilant 250 trail bike should fit within your $3000 budget! The money you will save on fuel will be a blessing to, I was getting 150km for $10.50 of fuel (@ roughly $1.40/litre)