How does progression work? What do you keep between games?

[Hot Dog Master]

I see people get items after games. Is it like Bloodborne blood gems? I was hoping you kept weapons you find, and can change your starting weapon around.

[BoomBoy]

They are relics. They boost your character in various ways.

[Solomon]

If it's like the network test, you get to choose between several random relics (basically perks) or an amount of currency. Some of them were very powerful, others were very lame, most were average. The perks basically help improve your build (max of 3 in the beta).
The shop where you can use the currency was not available in the beta.