fleece backed quilts are Wonderful!
you can choose a really lightweight batt, or a layer of flannel, or (for one quilt i did that had show-through) i used a layer of muslin instead of batting-it worked well....i would not use 2 layers of fleece- but i often use fleece for backs & i pretty much always either use a batt or a layer of muslin or flannel....works great.
as for direction- pull on the fleece in both directions- one direction will stretch more than the other- i generally load so the stretchiest is (side to side) with least stretch lengthwise. then just make sure you (smooth) as you go- don't stretch- makes fabulous baby/kid quilts!!!