There is a craft button inside the gear tab. You craft gear by taking 3 of the same gear type, and from that you can craft the next rarity up of that gear type. For example, if you take 3 common dual pistols, you can craft 1 uncommon dual pistol. It has to be the same gear. You can't take 3 random common weapons and get an uncommon weapon. It has to be 3 common shotguns, 3 common dual pistols, etc.

I have 3 of the same gear type that I can craft into better gear, but I've been leveling one up. Will I lose that progress?

No, when you craft a gear item, the level of the new, rarer crafted gear item will be equal to the highest-level gear item you used to craft. For example, if you have a level 20 uncommon dual pistol and two level 1 uncommon dual pistols, putting these together will craft you 1 level 20 rare dual pistol.

You can dismantle gear you do not want for coins and manuals. However, the more you hang onto, the more stuff you have to craft better gear. You might want epic dual pistols, but the game might give you drops that will net you an epic shotgun first.

When I kill enemies, sometimes the manuals that drop are different colors. What does that mean?

Up to level 30, upgrading gear requires only the regular black manuals and coins. After level 30, they require coins, regular manuals, and purple manuals.