-I got to level 20 last night and am confused as to what I should be doing now to continue to level up. Someone said I should be doing a bunch of Strikes and PVP matches hoping to get some equipment with Light in it?
There's tons of ways to get rare gear with Light, a full set of level 20 Rares upgraded will get you to light level 24. You can get those consistently as rewards from the level 24 Strike, and engrams will randomly decode into them as well.
-Is it possible for me to get a Legendary/Exotic loot drop at level 20?
Yes, very rare, and legendary engrams decode into useless shit 95% of the time. Don't get excited if you see a purple ball, you'll set yourself up for sadness.
Upshot, blue engrams can (very rarely) decode into a legendary.
-What do the factions you can join do? Should I join both or none of them?
They transfer all of your normal Vanguard and Crucible reptuation gains into that faction only - only join one if there's no rank 2/3 gear you want from the Vanguard or Crucible vendors.
You can just switch out the class item to change your rep gain back to normal though.
The benefit of joining them (beyond the rewards for hitting rank 3+ with any faction) is a set of fixed stat gear, you can get int/dis, dis/str or int/str from the different factions, plus they all have unique legendary weapons.
-Does basic EXP earned in battles mean nothing to me now, as leveling up is strictly done through picking up Light drops?
You earn a Mote of Light for each 'normal' xp level you gain. They're used as currency for a few things, notably 23 of them can buy you an Exotic Engram from Xur on the weekends (which, much like Legendary engrams, will troll the shit out of you and decode to another classes exotic, enjoy).
-Is doing Strikes from the playlist the same as going to a planet and doing Strikes that way?
No, you can't level up the 'normal' planetary strikes, presumably to keep everyone in the same matchmaking pool rather than farming one specific strike repeatedly.
Any insight to help me understand better would be appreciated. I'm with Ghaleon, none of this makes a lick of sense to me so far since I've gotten to 20.
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All that aside, as I posted earlier,
the 100% guaranteed, 0 RNG method of gearing up in Legendaries/Exotics is using Vanguard and Crucible Marks and Strange Coins.
Do your pve events for Vanguard marks, do your pvp for Crucible marks, turn in materials for more marks if you need to.
Strange Coins you'll get just by playing the game - they're rewarded for various activities and engrams can decode into them.
Consider drops/strike rewards/pvp rewards to be an added bonus with a chance of getting you extra legendaries/exotics, don't depend on them or you'll burn out and get bitter.
A single weeks worth of marks and coins will get you to light level 26, so you don't need to rely on the RNG to gear up.