There are three types of guns you can use, pistol, shotgun and rifle. Only your avatar can change their weapon category, all other characters have a set weapon which you can not change. You can choose different types of weapon within the category of the weapon though.
To change the weapon of your avatar, your avatar needs to have level two. Click on your avatar and choose between pistol, shotgun and rifle and then the type of weapon you would like.
To choose which type within a category of weapons you want to use for a character, click on the character, then on the weapons button and then choose your weapon.