Garter stitch is basically the name for a basic knit pattern. It just means you knit (stitch) every row, as appose to following another pattern where you do more then simply knit stitch.
The large 10's work well because you can really see your stitches, however you should use the needle size recommended for the yarn on the label.
Make sure the way you're knitting is the most comfortable way for you. I had a hard time learning until I discovered I'm a left-handed knitter :P. I also found it easier if you turn your needles to make a right angel, that way you won't have to worry about the yarn slipping off by accident.