Thread weight makes more of a difference than color. Invisible monofilament thread like Superior's Monopoly (a polyester), Filtec Essence (Nylon) and YLI Invisible thread (also a nylon)would do what you are asking but they can be fussy and lead to frustration. Do experiment with them because it would benefit you to be able to take advantage of these threads when the need arises. Edited to add, some people have found they need to use a thread net with these threads. I do when I use the big cones on my LA but I have a tiny cone for domestic machine use that I put on my large cone holder for my domestic machine and have had no issues. I believe that one is YLI. I use Essence and Monopoly on my longarm. All monofilament threads do have a certain amount of shine to them and are not 100% invisible.
Threads that are more user friendly and don't show much (texture rather that quilting stitches) are super fine threads like Wonderfil brand Invisifil which is a 100 wt polyester. Superior also makes one called Microquilter. I prefer Invisifil but that is just me. Here is an example of a quilt done with all Invisifil.
Batik block swap quilt
Silk thread will also show texture. Just google images of Diane Gaudynski to see examples.
As far as color, well that all depends on the value and colors in the quilt. If you use a lot of high contrast value (light and dark) or brights there will be places where the thread will show more than other places no matter what you use. It has been my experience that sage green, a greenish gray and golden tan seem to play nicely with everything.