Rifts are multi-level dungeons where you focus on killing enemies until the rift guardian is summoned. Killing elites yields faster summoning progress, and killing the rift guardian yields good loot, gold and exp. This is generally a good place to level and farm. Occasionally, Rift Guardians will drop a rift key, which will let you open a Greater Rift. Greater Rifts function nearly identically to normal rifts, with a few exceptions: You can choose a difficulty, you have 15 minutes to complete it (defeat the guardian), and no enemies but the guardian have drops. After clearing a greater rift, your time is entered on the ladder, and you get to choose legendary gems to upgrade.

Rifts are generally the best place to farm. Just pick a difficulty you can clear comfortably, and you'll eventually have good enough drops to move up, wash, rinse, repeat.

Adventure Mode is story mode without the story, unlocked after completing all five acts with any character. You are free to roam through the five acts, and there are a handful of new areas to explore. Each act has 5 bounties, which are basically "go kill this guy/ go do this" missions. Completing all of the bounties in an act yields a cache, which can be opened for act-unique crafting materials (useful for the cube, blacksmith/jeweler plans, gold, and some legendary loot (certain items are cache only, and found only in specific act caches). If an Act is marked with a bonus, a second cache will be awarded upon completion. Keywardens are uniques that have the chance to drop infernal machines, which you can use to ultimately farm materials for hellfire rings and amulets.

Wherever, just join a game you feel comfortable clearing. Or make one of your own. Chances are, there are plenty of people looking to farm that are also in a similar position as you.