100% cotton thread is usually preferred, but a lot of people use cotton-poly. The idea with 100% cotton is that it will not "saw" through the 100% cotton fabric. Polyester is stronger than cotton so, with lots of wear, polyester thread can eventually cut through cotton fabric. Poly-cotton thread is typically cotton-covered polyester so the cotton coating can wear off.
You would never use hand quilting thread for piecing. It is made heavy and stiff so the thread will not knot while quilting. It is too heavy and stiff for piecing.
Since you mentioned basting, are you using paper pieces? And intending to whip-stitch the pieces together? For that the heaviest thread you would want to use would be a normal weight (50/3). I would be inclined to use a finer weight thread for whipstitching (60/2) so the thread doesn't show quite as much. I'm not an expert on whipstitching, though.