Someone might have already posted this link, but its such a good site it won't hurt to double up.
I just spent a good 2 hours googling GARTER STITCH GRAFTING and finding conflicting instructions on where you knit and where you purl. I tried over and over and had to keep unpicking my work. It wasn't the actual concept of using the tapestry needle to stitch my knitted live stitches that was the problem, it was the fact that there were so many different instructions for the one procedure and they weren't working.
I finally found the above site, which covers stocking stitch grafting and reverse stockinette grafting, as well as garter stitch grafting. What turned on the lights for me was her mention that some websites mistakenly call reverse stockinette grafting "garter stitch grafting" .... and I'd obviously found some of them! Thank goodness, it wasn't my stupidity after all!
So I followed the instructions given by Woolly Wormhead, and 5 minutes later my graft was done, perfectly.
Her tutorial is worth printing out and tucking away till you might possibly need it. It might save you 2 hours of frustration like I just went through!