Maybe you could follow the directions that you find online but just put new fabric on top w/o striping off the old, replacing padding, and retying the springs. Other than complicated shapes (which is the catagory wing chairs fall into), it is very similar to stretching a canvas on a frame if you have ever taken a painting class. Center the pattern of the fabric on the chair, staple a few staples in the middle of each side, pulling it taut as you go, and work your way to the corners on all sides simultaneously (don't start at one side and just work your way around, kwim?).
If your staples are someplace that shows, you can cover them with that flat decorative trim and nail upholstery tacks on top. You can also fold the fabric under and handsew with a simple running stitch if you need to attach fabric where there is no wood to staple into.
The easiest chairs to reupholster are ones that just have a fabric covered seat (like dining room chairs). Usually the seat unscrews and comes completely off. Then you staple new fabric on and screw it back on.
Mom to unschooling 4everboy since 8/01