I heavily starch the binding fabric before cutting it into strips, using a 1:1 solution of Sta-Flo liquid laundry starch and water. This stabilizes the binding strips so they don't distort while sewing.
As someone else mentioned, I also use a walking foot.
For the quilt itself, I use a different method than most people. I mark the finished edge of the quilt with a Sharpie; I do not cut the edge! To sew on the binding, I line up the cut edges of the binding with the Sharpie line to sew. For mitering the corners, I treat the Sharpie line exactly as if it were already cut. This method leaves something to "grab on to" at the right side of the needle. If the quilt is especially puffy or difficult to handle, I will machine baste outside the Sharpie line to make sure the quilt is stabilized while sewing on the binding. To minimize bulk, I will often trim the quilt edges -- but always outside the Sharpie line so I still have some quilt on the right side of the presser foot when sewing on the binding. The only thing to be ***very*** careful about is when you finally trim your quilt edge. (Incidentally, this method also allows you to adjust how much quilt is inside the binding for fullness.) At the corners, do ***not*** trim off any of the binding. If you want to trim off the quilt sandwich corner, you can do that, but you *cannot* trim the binding (unless you really want a hole in your binding at the corner when you turn it!).