I think this is one of those questions where there isn't a single "right" answer as to what is "best". It depends on how you're paper-piecing and what your preferences are. And as you gain experience and develop and adapt your technique, your preferences will change.

I started out with copy paper, then moved to newsprint, then freezer paper, and now I only use vellum, because that's what works best with my technique.

I no longer sew through the paper. I use a single little dab of glue stick to stick the vellum to the fabric. Then I fold the paper right on the line (and this is where vellum comes in handy, because you can see the line through the paper) and I stitch right next to the line. When I'm done with the block, I pull the vellum pattern off the fabric and reuse it. Saves paper, and I don't have to deal with ripping it and cleaning up all those little pieces.

I use the cheapest copy paper that I can find. I have also purchased a ream of newsprint 8.5 x 11 and used that as well.