Through the campaign? Please forgive me but I did not play the beta.
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1. Kill shit. 2. Keep an eye out for green orbs, blue glitter, and generally anything that pops out of an enemy when they die. 3. Chest. There are chest hiding around the game that provide you with all kinds of goodies. 4. Completing levels, strikes, public events, etc. 5. Completing task such as bounties. Bounties are what you would call challenges... sort of. You go talk to this guy at the Tower and he has multiple bounties you can choose to complete. Some are for Crucible only. These can have you doing anything from getting X amount of melee kills without dying to killing X amount of this type of enemy. You can carry up to 5 bounties. They change daily. You have to return to the bounty guy once you have completed the bounty to collect your reward. And of that was all in the beta. Also, you can work for certain factions that also give you gear (loot). There's no telling what else they have in store for us on release.