Weapons are upgraded through vending machines. The Minuteman's Armory will allow you to upgrade your weapons. The weapon has 4 upgrade slots, and a few of them even have the first unlock already in place. One of the best weapons in the game is the Carbine, so focus on upgrading that since you are limited on money (especially in 1999 mode). Vigors only have 2 upgrades and can be quite expensive.
The cheapest full weapon upgrade is the starting pistol.
The cheapest Vigor to upgrade is Bucking Bronco.