You might give your date the garter during the wedding reception. If you catch the garter that the groom has thrown, you will then be encouraged to put the garter on the leg of the woman who caught the bouquet.
For the girl (garter are for girls normally), a garter goes on the left leg. But whichever you think is best. No rules for garter really. But most girls nowadays wear those thing that you put on the wrist (and usually with a flower design on it).