Weapons technology seems to be one of the dominant skills that influence your damage abilities. Anatomy also affects your damage abilities too. But I have a bit of a problem with my skills. I have only ever used Laser or BLP, no melee and not a whole lot of explosive weaponry. So... My damage skills are 5 levels apart from hit skills in Laser, and 2 levels apart in BLP. It seems odd to have such a large gap...
But I used to hunt things I could kill without needing to fap. And not fapping means I am not gaining Anatomy. And not gaining Anatomy means my Damage skills are hurting. But if you don't want to be a medic or waste fap trying to skill up, then there is one more option.
If you are wanting to use Laser, craft some laser weapons.
If you need to increase your BLP damage, craft some BLP weapons.
That's really the only ways I know of getting damage skill up. But Anatomy is decently cheap to chip if you decide to go that route.