The biggest way to increase your damage done is to aim for the critical location on enemies. You will increase your damage output immensely by constantly hitting those spots.
Of course this also requires you to learn where those critical locations are. This would be a very large list for every specific enemy in the game.
For most targets, the head is a safe bet as the critical hit location. Exceptions are spiderants where their exposed tail, varkids have a small eye like area on their rear section as well, and all robot types (they have 2, the lighted "eye" is almost always a spot and some you can shot off their limb joints as well. The joints aren't always a spot as some robots are designed differently than others.
If you are playing Zero, the assassin, you can invest into B0re skill. With it, every time you use his special ability, the critical hit locations of the enemies will be highlighted with a red hexagon.
Also consider your class talent choices. Each class has talents that increase damage done by a generic or specific gun type. Those talents will further increase your damage done to targets.
Make sure you keep upgrading your guns as well (replacing older guns with new, more powerful ones). The base damage they do directly impacts how much damage output you have. Elemental Effects play a huge part as well. Please make sure you read this entire answer.
Different weapons also have different multipliers for these critical hits as found on a table listed here.
The list below is listed as Weapon Type -> Critical Damage Bonus
- Assault Rifles -> +66%
- Launchers -> no criticals
- Pistols -> +100%
- Shotguns -> +100%
- Sniper Rifles -> +300%
- Sub-Machine guns -> +100%
Jakobs weapons are special:
- Assault Rifles -> +130%
- Pistols -> +150%
- Shotguns -> +130%
- Sniper Rifles -> +420%!