I use the cheapest aka the thinnest muslin I can find. Wash and dry to shrink, cut to needed size and stitch away. The thin muslin does not add significant weight, and (more important to me), does not have to be removed. The last I bought was from Fabric.com and I think it was around $2.00 a yard....cheap.
As for quilting finished quilt, I generally just channel stitch or in the ditch around blocks. For my next one, I think practicing free motion and domestic ruler quilting would be perfect on a string quilt. Mistakes won't be so obvious.