Wildstar has several types of currency, and this is a quick guide on each one.
First, there's the "standard" copper, silver, and gold. These are used for most stuff that you pay for in the game.
Renown is earned by doing just about anything in a group--whether it be questing, PVP, dungeons, raids, etc., all of these things will slowly earn you Renown.
To spend Renown, just go to a Renown vendor (they're near the Guild Services guys, just ask a guard in town to find them).
Most of the stuff you can buy with Renown has to do with character customization, things like dyes, mount customizations, house items, but there are some things you can buy that will affect game play too: consumables. The effects these consumables have vary, so go have a look. You can expect things that increase the rate at which you gain experience, crafting packs, and more.
The coolest thing you can buy from Renown vendors, at least in my opinion, are hoverboards. Hoverboards are a type of mount, and they are too much fun to use for getting around. If you do a lot of grouping up to the level where you can buy a mount, you can expect to be able to buy your mount using Renown alone, and not have to spend your gold.
Prestige is the PVP currency. You earn Prestige by, you guessed it, PVPing.
Battlegrounds and Arenas are the best way to earn Prestige, and you'll be able to buy PVP gear from Prestige vendors using your Prestige points. The gear that you can buy is level-appropriate Battlegrounds gear, as well as level-cap Arena gear.
Something to be aware of is that Prestige is capped, meaning you can only have 5000 Prestige amassed at any time. The reason for this is so that you go spend it and keep your PVP gear as good as it can get.
Wildstar Elder Gems
Once you hit level-cap (50), you stop gaining experience and begin accumulating a currency known as Elder Gems. These gems can be used to purchase items from Elder Gem vendors.