I do like that tutorial for a beginner. Even if you just make a table runner or pillow cover, it's enough to get the basic idea of paper piecing across.
I actually prefer to not stitch through the paper. There's a step in that tutorial where they have you fold back the paper on the cutting line, line up the add-a-quarter ruler and trim the fabric, then align your next fabric patch and stitch on the line. I leave the paper folded back, align my next fabric patch, and stitch right next to the fold in the paper. That way I don't have to tear all the little pieces of paper away.