Hello fellow players:
I am a somewhat new player, wondering how/where I can buy new clothing? I’ve looked around the trade district, but I have not been able to find anything that I could really “wear” for clothing (unless I completely missed something).
Thanks!
Welcome to WoW!
You can find gear on the Auction House. A lot of times though people will put gear in there for over the amount a new player could afford. You’re better off getting new gear by doing quests or dungeons. It never hurts to look though.
Oh, and keep an eye out for chests whenever you’re questing in the open world. They usually always have a piece of gear inside them.
Follow up through quests, they usually give armor cosmetics,
Welcome to WoW!
Don’t spend gold on clothing / weapons, unless you want it for RP purposes. You outlevel the gear you get so quickly as you level up, it’ll be wasted gold.
The blues & greens you get from questing and dungeons should be all you need until you hit max level. Just keep replacing your gear as you get better items.
Welcome to WoW. I hope you’re enjoying the game 
EDIT TO ADD: It occurred to me that you may not understand the meaning of the term “blues & greens”. It’s a reference the colour of the name of the item. Items have colour-coded names based on their quality (when compared to other items at the same level).
Welcome to WoW.
The Auction House will have all the “clothes” you need. Be sure it’s the same armor type that your class wears. Priests wear cloth type armor.
If you find the stuff you want to be more than you can afford, do quests. You’ll often get the stuff you need through them.
Also, if you havent already, theres a quest in the Eastvale Logging Camp (eastern Elwynn forest) that will give you a Red Linen Shirt as part of the reward for doing the quest.
Tailoring may also be beneficial for your Priest. You can make your own clothes with Tailoring, with the cloth that drops from humanoids.
It is probably worth mentioning what other game you are coming from.
WoW doesn’t really have traditional “cosmetic” options.
Your look is based on your gear. You can transmogify (i.e. change appearance) to other looks you have acquired as gear previously).
There is no real concept of “clothes” or “appearance only” outside of current/former equipment appearances - with a few exceptions you will pick up on over time (but they are quite few).
Welcome.
Questing is probably your best route like Hella said.
Another option is to become a tailor, it allows you to make armor.
Buying armor as you level is a bit of a waste since you will level so fast you will out grow it so to speak rather quickly.
Anyway welcome again and I hope you have a lot of fun in your adventures.
Welcome Throellin!
I know it might sound a bit unintuitive, but most NPC armour shops in WoW don’t sell anything that’s actually good for your character (Blue quality or higher).
You get new gear from doing quests, completing dungeons, playing battlegrounds, or (like other players have mentioned above) you can purchase it from another player off the Auction House.