TLDR: Gear recommendations for a level 5-6 Barbarian

Just started a campaign with some friends (first game in almost 20 years)

Playing a Level 5 Path Of the Giant Barbarian.

https://www.dndbeyond.com/characters/106176365

My DM gave me a +1 Great Axe and a Belt of Hill Giant Strength based on my backstory and how I was worked in later in the campaign.

Got some gold from the first quest, and a fairly good roll for starting gold giving me about 640 G.

What should I prioritize purchasing on my first trip into town? I was thinking about prioritizing items to raise my AC, as its pretty low right now, but I am open to suggestions!

Thanks!

You could try to buy a medium  Armor, +1 to increase your AC, either a scalemail, breastplate or half-plate.

Thanks Half Plate was definitely on my list!

Just remember when you're wearing armor some of your features don't work: Fast Movement and Unarmored Defence (obviously)

If you do want that extra movement you could go for some Bracers of Defense or something that gives you extra mobility.

Fast Movement

Starting at 5th level, your speed increases by 10 feet while you aren’t wearing heavy armor.

Stay in medium armor to keep fast movement working.  Can't rage in heavy armor anyway.

Alternatively, adamantine armor is good on a barbarian, makes your high HP pool stretch further!

ring or cloak of protection, or bracers of defense... assuming that you can buy magic items.

Boosting your armor class is good, but boosting your saving throws can be more helpful in the long run.  Items that grant resistance to elemental damage are also pretty useful, like Boots of the Winterland.