Make Do and Mend

To upgrade your weapons you first need the Gunsmith skill - for which you'll need the "Master Crafting Manual" and at least 7 skill points. See the linked Gamespot guide for where to find it.

Make sure to collect all Gun Oil and Scrap as you're exploring, so that you then have the mats to craft the upgrades once you've got the skill.

Its worth upgrading the revolver, to make getting Target Practice easier.