The standard way of getting firearms in DL2 is as follows:


Reach Region 2 → start the “Lost armory” quest → get the 1st pistol
Buy more firearms at Jai’s after completing the quest (getting to the quest (region 2) takes 6-7 hours)

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How to obtain firearms in DL2 during the FREE WEEKEND?


You can complete 2 COMBAT CHALLENGES focused on firearms called Firearms Mayhem and Quarantine Standoff in Region 1.
Their locations are: the main PK terminal station & the THV GENOMICS CENTER next to Horseshoe Water Tower 


The above challenges stay in the game permanently, even after the free weekend is over.


Another way is to purchase the Volkatronix Redux Rifle+100 bullets at the in-game store.