World Tier System
World Tier selection (changed on your megamap) is a brand new feature for 1.4, and if you just hit level 30 it's now relevant to you. The reason this is important is that the World Tier (WT) corresponds to the quality (gear score) of loot you can receive. WT1 drops 163 gear and weapons & and power level 30 mods, WT2 drops 182 gear and weapons & power level 31 mods, WT3 = 204 & 32, and WT4 = 229 & 33. The difficulty of the content you do dictates the quantity of loot you receive. Once you reach a certain Gear Score threshold, you will be able to (and you will want to) increase the WT so you start getting better drops. Tiers 1-3 do have a low chance of dropping items from the higher tier, but you won't need these rare drops to "tier up."
World Tier also dictates the quality (and price) of the items sold by vendors. Yes, ALL vendors are relevant again. Not only do the safehouse vendors sell HE weapons, gear, and mods, but also caches that give you random items at that WT's gear score plus a handful of Phoenix Credits (PxC). If you're ever at a loss for what to farm, just plain credits will open the door to a lot of options. Check the Base of Operations (BoO) vendors daily. Also, downstairs from the BoO are the Phoenix Credit vendors, and they are relevant now too. One sells HE weapons and gear as well as Gear Set items at the gear score of your World Tier, but he also sells more specialized caches for 100 PxC each. The other vendor sells blueprints (including some top tier ones) if you're into crafting. Their stock changes weekly (Friday evening at 20:00 US Eastern Time).
Once you've set your WT, you can literally do any content in the game to potentially get any weapon, gear piece, or mod at the gear score you need. You can potentially walk down the street, kill a random roaming Rioter (aka "Alex"), and get a Gear Set piece as a drop. If you want more quantity of loot, though, you have to do more difficult content. Like what?
Well, Hard Mode story missions are lucrative and easily soloable. Challenge Mode story missions are obviously more difficult but provide higher rewards. Whether or not you can actually complete them solo is kind of another story, but it's quite common to use Matchmaking to group up for these since they don't require a ton of coordination (unlike Incursions). Both of these are initiated from your megamap or from directly outside the mission itself (just like Normal Mode story missions for you fresh 30's). Not only will you get a guaranteed mission completion reward (items + credits + PxC) and boss drops (items + PxC), keep in mind that every single enemy can drop relevant loot. Sure, you'll see some purples, but you will also see golds & greens.
Bullet King is back! For those of you who don't know what that means, there are named enemies in the open world (aka Light Zone) who drop guaranteed gold and/or green loot. They respawn every 4 hours. Yes, you can also kill them on your alt characters. Very soloable.
High Value Targets
Also good are High Value Targets (HVTs). Vendors for these are found in the BoO, in the Security Wing near the Scavenging Crate, or in the HUB (downstairs from the BoO) near the restock & stash boxes. HVTs are an unscripted encounter against some waves of enemies and at least one named enemy (boss). To purchase an HVT contract, you need intel, which you get from doing Search & Destroy (S&D) missions at safe houses. The best ones to do are "critical" S&D missions, and the safe house will have a spiky icon to let you know it's S&D is "critical."
You will need to clear a neighborhood of all Side Missions and Encounters before you can do S&D missions there. You initiate them by interacting with the big map inside the safe house. Keep in mind that the enemies you kill to get intel can also drop loot! Once you have enough intel, purchase a contract (you can only have one at a time), and go kill the target(s) for big rewards. HVTs are divided into Daily and Weekly (two different vendors) which you can pretty much view the same as Hard and Challenge mode equivalents, or as I refer to them "easily soloable" and "holy shit 5 gold shotgunners are running straight at me." THESE NO LONGER ONLY DROP LONESTAR GEAR! Just like with roaming the open world and hard/challenge missions, any enemy can drop any item (depending on your WT).
Not enough options for you? Well, Incursions are group-focused content centered around specific objectives. All incursions are tuned for a 4 player team, so soloing them, while technically possible (except Dragon's Nest I guess?), isn't really lucrative or efficient. Remember, higher difficulty = more loot. The incentive to do Incursions is they drop more Gear Set (green) items, and each Incursion is weighted toward dropping certain gear slots (formerly weighted toward dropping certain sets). Falcon Lost (located near Warrengate), which now has checkpoints after each C4 explosion, drops Gloves and Masks. Clear Sky (located near Autumn's Hope safehouse) drops Chests and Holsters. Dragon's Nest (only available with the Underground DLC) drops Backpacks and Kneepads. Matchmaking for incursions does work, but due to the increased difficulty and need for coordination, having a mic is recommended.
Unlike other content in the game, however, non-named enemies in Incursions do NOT drop loot. This is because Incursions have infinitely spawning enemies (either under certain conditions or just all the time).
Underground (requires Underground DLC)
The closest thing to "dungeons" that this game has, the Underground (UG) is an objective-based DLC where the encounters are relatively random. There's a lot to it, but think of UG as the equivalent of story mode missions that aren't exactly the same every time you run them. Plus, there are modifiers called Directives that you can add to each UG mission to increase the difficulty (and thereby increase the loot quantity PxC reward, thanks /u/Kamoi) such as not being able to use your minimap or losing all of the bullets in your magazine when you do a partial reload. UG missions are soloable (YMMV) but you can also use Matchmaking here to increase your tactical options. The Underground Terminal area also has a PxC vendor with a variety of items (also on a weekly reset).
I mentioned vendor caches and PxC vendor caches, but there are many other types of caches in the game now. Completing daily/weekly Assignments (found on your megamap) will merit you caches, getting 400k XP will merit you a cache, ranking up in the Underground (past rank 40) will get you a cache, and you can also pick up caches in the Dark Zone (DZ). So, the entire time you're killing stuff and getting loot, you are also getting caches that will give you loot. And yes the "Increased Kill XP %" stat and the "Triple Accolades" talent that rolls on HE Kneepads do get you caches faster! You'll see when you do a story mission or HVT or when you kill an enemy, you get a varying amount of XP so just by doing content you're getting drops and caches. Everything you do has the potential to give you loot.
The DZ is the most farmable, most dynamic, and most dangerous part of the game (and probably the reason at least some of you returning players left in the first place). THIS HAS NOT CHANGED. Currently you are just as likely to get shot in the face by another player as you are of finding those "god roll" Sentry's Call Gloves that you really need to finish off your build. Because it's such a dangerous and difficult place, you may notice you get more loot drops in the DZ, but because you also have to extract them to be able to use them it balances things out (and requires you to be more choosey about what you pick up).
The big thing to mention here is that the 1.4 patch changed the B.L.I.N.D. set into the Banshee set, which changes things in the DZ slightly. Long story short, the 2 piece bonus for the Banshee set is an increase in DZ currency that enemies drop, while the 4 piece bonus makes it so you do not lose DZ XP or DZ currency when you die (as long as you're not a Rogue). It does some other things, but I mention this specifically because it allows for farming credits and DZ XP without losing your character's progress (you'll still drop contaminated items you're carrying if you get killed by a player or NPC). To me, this means the DZ can be fun because it takes away most of the consequences of getting cheaply killed. There's some controversy around it, and around PvP in general in this game, but the fact remains that non-PvPers can now get some use out of the DZ if they choose to.