I've used Coat and Clark cotton covered poly for piecing for the last 25 years. My old quilts that I wash in the washer/dryer all the time are doing GREAT. It's harder to get that thread, so I switched to poly covered poly. It's fine too. The important thing for me is to use 37 weight thread at least. Lots of people use 50 weight and finer, and I just don't think that thread is really strong enough. But that's just me.
For quilting, I used Coat's or Star cotton, but more recently have gotten more creative and have been using Glide and Mettler trilobal polyester and Signature cotton.
I hear lots of talk about linty thread. The lintiest part of quilting is batting, followed closely by fabric. I've found that white on whites tend to be especially linty. Thread falls last in that spectrum. Look at the color of your lint. You'll see why I say fabric is a huge contributor.