Cone thread is different from serger thread. Cone thread is usually 50 weight if cotton, but could be something else. 50 weight/3 ply cotton is always a good choice for piecing and can be good for quilting. Serger thread is two ply and not intended to be strong. That's why you use at least two spools of serger thread per seam, often more, plus a regular thread in the serger's needle. I would never use serger thread for anything except serged seams, and I did once serge two quilts, all seams, plus use a 50/3 cotton in two needles, and both quilts are still doing just fine 20 years later. Yes, serger thread does come on a cone, but that's where the similarities end.
40 weight cotton, normally 3 ply, is what is used for quilting most often. That's where you find your variegated threads. It is a stronger, thicker thread that will stand up to tugging on a quilt. There is also 30 weight, 35 weight, 12 weight, and others, which are used for decorative impact. The lower the number, the thicker the thread. 12 weight thread is intended to lay on top the fabric and be noticed. 60 weight thread, which is often polyester, is thinner than 50 weight. It could be stronger than 50 weight cotton in some threads, except for spun polyester, which is made of short pieces of leftover poly put together. I might piece homeless quilts with spun poly but I wouldn't quilt with it.
Which brings us to a controversy that's been going for many years: can you use poly thread on cotton fabric? Some say the thread will eventually cut through the quilt. Indeed, I've seen several examples of that happening. Superior threads says it is okay with their threads. I've used Superior and have several times had long talks with their representatives and would be inclined to trust them, but I would still test on a smaller quilt before doing an heirloom quilt with poly.
Rayon thread is decorative. It is not strong enough to be a primary quilting thread, IMHO, and definitely is not intended for piecing. It is good for thread painting.