How to increase GS to unlock HC dungeons?

I'm new to the server and got my first max level. Been grinding some normals but I'm trying to get into the harder 5mans (normal ToC/PoS/foS/HoR) and heroics but I'm stuck gear score wise and don't know how I'm supposed to overcome it.
I'm currently 70% geared from normals (2.7k gs), so the chance I get an upgrade from them is very slim, yet I don't know how else I'm supposed to get to the required breakpoints to unlock the other dungeons.
Since I'm new, buying BoEs off the AH isn't really an option considering their price. What am I doing wrong?
I thought having 70% gear from normals would be enough to unlock at least something new but it ain't so.

Farm fos/pos, buy stuff with emblems of triumph.

No one seems to want someone with my GS for fos/pos, and I can't get emblems beyond the 2 from the daily...

Only queue for high level normals: (Oculus, CoS, Utgarde, Halls), so you won't do The Nexus over and over.
Also, wear a faction tabard. There's a chance you'll get some rep from instance mobs. This helps a lot especially if you're farming rep by questing too.

As soon as you get 193 ilvl, you're done.

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While you're gearing and hitting those upgrades, relog. The game doesn't seem to update your avg item level that fast, if at all for the queue requirement. So after getting gear, relog and then try queue. If you still can't queue, find more upgrades, and repeat.

semi-cheap blue items from lvl 78 pvp sets, BoE ( some cheap some not so much ), you can use vote shop for hard to get weapon/trinkets.