When you tap the Equipment button on the main menu, you are brought to the Equipment screen. On this screen, you can view the stats of your currently equipped equipment on the right side of the screen, and you are presented with various gear categories at the bottom of the screen.
You can choose which mode to change equipment for. Standard indicates PvP mode equipment, and Zombie indicates Zombie Event equipment.
When you tap on a piece of gear that you don't own, you will be provided with options to purchase them in the bottom-right corner.
If you tap on a piece of gear that you own and haven't got equipped, you will be shown the "Upgrade" and "Equip" buttons in the bottom-right corner. Tapping on "Upgrade" or selecting an equipped piece of gear will provide you some purchase options to improve it further. Note that some pieces of gear require you to have purchased and fully upgraded a lower tier piece of gear before it can be purchased. This will be indicated by a grey button in the bottom-right.
As you upgrade your equipment from here, you may see the text "Overload" appear above the payment options. This appears once you have upgraded the equipment to its limit for Campaign mode. Any further upgrades will only apply to PvP for standard equipment (overloading event equipment works like normal upgrades except it focuses on one stat).
Finally, you can change your skin tone and gender by tapping the icon just below the back button in the top-left corner.