Beating campaign and reaching level 70 is in many ways the start of the game. Diablo 3 is (in its current state) really end-game focused. Typically people, after beating the story for the first time, don't touch the story mode.
Adventure mode is where it's at. There are many activities, such as Bounties (Different objectives around the world, marked with an exclamation mark in the map). Completing all 5 bounties in a single Act gives you a bag with crafting materials and items. Some meaningful legendaries are found this way. Nephalem- and Greater Rifts are the main content though, and the most efficient way of obtaining legendary and set items.
Nephalem rifts are pretty straightforward: open the rift with the obelisk in town, kill enough monsters to summon the Rift Guardian and kill it for loot. Nephalem Rift Guardians drop Greater Rift Keystones, which are used to open a Greater rift.
Greater rifts are much like Nephalem Rifts, but they don't scale with game difficulty. Instead they have their own level/tier (chosen when opening a Greater Rift). Gold or loot doesn't drop from monsters in Greater Rifts, instead all of the loot is dropped by the Rift Guardian. Additionally you get one Legendary Gem from each Greater Rift you complete, until you have one of all Legendary Gems. You can upgrade these gems if you kill the Rift Guardian within 15 minutes of starting the Greater rift, making them stronger.
Clearing Greater Rifts on a level where you can complete them in 5 minutes or less is the most efficient way to get legendary or set items.
One really important aspect of adventure mode, and end game, is the Kanai's Cube, which is found in Elder Sanctum (Inside Ruins of Sescheron of Act III). It has plethora of different uses, check it out in the wiki or discover them yourself after finding it.
The next big goal for you now is to gather a complete 6-piece class set (green items). Having a full class set equipped is an astronomical boost in power, and they define your playstyle. In patch 2.6.6 there are 4 6-piece sets for each of the classes, supporting different kinds of skills and playstyles.
Edit: If you are playing on console, some of the things I said might not be true. Diablo III is a bit different on console, and I have not played it.
I don't know why I took the time to type this, as all of this information is easily accessible and probably better explained in various places in the internet. I was bored, and wanted to be helpful, I guess.