I just answered this in your last question! :) I like tracing paper...it's cheap and it rips well without having to perforate the lines ahead of time.
I would not use newspaper, for a myriad of reasons...but the most damaging I would think to be the ink getting all over your machine and your fabric and making a huge mess. If you're on a budget computer printer paper or tracing paper work fine. If you use the heavier computer paper, just sew along the lines with out any thread in your sewing machine before you start paper piecing, so it will be pre-perforated and rip easy when you're ready to take it out. With tracing paper you can eliminate the pre-perforation step, because you will be using small stitches for paper piecing and that's enough to loosen up your paper.
On the other threads many of the ladies like the specialty paper you can buy that is made specifically for paper piecing. I don't find it necessary for my needs, but I can see how a specialty paper of that sort, say, if it is water soluable, would be very helpful to someone who might be selling their quilt, putting it in a show, maybe making the quilt for a baby, or just want perfect results. :)