You will have to upgrade any weapon twice within the same category. Each weapon you get has two upgrades per effect. But you will have to wait till a certain part later in the game to be able to upgrade your weapon, the game will let you know when you have unlocked upgrade level 2 for your weapons.
You will have to upgrade a weapon to level 1 first, before you can upgrade it to level 2. You can choose what ever weapon your prefer to use. Id recommend either upgrading the shotgun or assault rifle, with fire (the fire will kill the zombies quickly). But its up to you, any weapon with level 2 upgrade will kick ass.