A step up from a rag quilt would be a faux cathedral windows. You could make it the same way as the rag quilt, then just sew down the circles on the top side. A straight stitch would allow them to rag, and a zigzag would hold them down.
My grandmother started me with a nine patch (a 4 patch would work as well), and a grandmother's flower garden (hand piecing).
I'm sure you'll see lots of other ideas.