I just re-paste this whenever it gets asked. Great strategy, works wonders.
Grey - toss up, I usually sell for extra currency as they only earn base secondaries with no upgrades until sliced. The slicing material is better used on a green mod. Diamonds, circles and squares will fall in buckets. Sell away, however; always keep +speed primary arrows, and LOCK them as soon as you earn one. I also keep crosses with potency, offense, tenacity, as well as Crit Chance/Crit Damage triangles.
Green - upgrade to level 9 to see if you roll +speed on any of the four secondaries. If you did, go to level 12. If it hits on speed, fantastic, if it hits on a character specific stat you need (offense for attacker, potency, +protection etc) almost as good. Level that mod to 15 and use it. Slice only mods that upgrade twice (2) on speed.
Blue - upgrade to level 6 to see if you roll +speed on remaining two secondaries if it wasn't there to start. If no speed, sell the mod unless it has great stats you might need for a "B" or "C" team. If it shows speed go to 9 & hope for an increase to speed. If not, that's ok, go to 12. If it didn't hit there you still have a +3-6 speed mod that is ok for a low-use toon. Don't slice. If you hit (2) or (3) on speed then try slicing it to purple.
Purple - I tend to keep most purples unless they have absolutely garbage secondaries. Might have one of the first 3 secondaries as speed, if not; increase to 3. If you get +speed keep upgrading. Ideally you'll strike a (4) upgrade on speed, which is pretty rare. But I have plenty of purple mods that have several (2) stats and +8-10 speed. Decent mods but I won't slice. Only slice if it's (3) or (4) on speed. Not worth the larger investment of slicing material for less than that.
I'll add as well that I typically go through once a week or so and sell any green blue or purple that have no speed secondaries whatsoever. I'll occasionally keep one if it has a ridiculous secondary stat for offense or protection etc.