If you're going to be woodworking, you need to sharpen tools, there's no way to get around that. As far as chisel go, $50 should be enough to get you started. 20 for a set of decent enough chisels, $1.50 for a few porcelain tiles, $5 for some wet dry paper. Wanna get really fancy, spend an extra $20 and get a sharpening jig.
It's a pretty common misconception that you have to have a couple hundred dollars in various whetstones, diamond stones, platinum stones. Arkansas stones, blade guides, blade guide guides, milling machines to flatten the guide on the blade guise guide, etc just to sharpen a chisel. Any abrasive harder than the steel will sharpen the blade. As mentioned, I like sandpaper. In a pinch, a cinderblock will work