Yes, use fleece! Especially for baby and lap quilts - the 60" width is usually sufficiently wide for them.
I used a single piece of fleece only in one quilt - it was both backing and batting. It was very soft and snuggly. I do worry about pilling over the long term when I do that, though, so usually use fleece for batting and put a cotton backing on a quilt, too.
I get the lighter weight fleece at Walmart to use for baby quilts. It's light weight in terms of how heavy it is as well as how warm it is. I figure that babies aren't so strong that they'll want to drag around a heavy quilt!
I've used the heavy weight fleece, too. It gives a quilt a soft, pillowy feel.