Destiny is a complicated game, and it does a fairly poor job of explaining its more advanced systems to the players.
Fortunately, the community has come together since release to compile a very useful list of third party services that help navigate the game's murky waters. These sites will support your Guardians in a variety of objectives, from public events to loot tracking.
A database for almost anything you could need to know, Destiny DB is a great general resource. In particular its item library is very useful for checking out all types of gear across rarity tiers, with detailed 3D models and full statistics and talents for each item. There's an impressive amount of other features, including info on Grimoire entries, all of the vendors, and guides for just about anything else in the game.
An extremely useful site for tracking your character progress in weekly activities and current status. By typing in your own Gamertag or PSNID you can get individual info panels for each of your Guardians with their inventory and ranks. Most importantly, it displays the weekly progress, keeping track of which Strikes or Raids you've completed on which characters. It does this much more effectively than anything in the game itself, and can keep you from forgetting which character already ran Crota's End or the Nightfall for the week.
Whether it's to complete a bounty or you need the drops on offer, players are usually in search of any ongoing public events. They seemed randomized, but they are on timers and the site linked above can track when they'll occur on each planet. Simply go on the site and pick which planet you'd like to check--all of the impending times are listed. They're pretty accurate, but sometimes it's off by a couple minutes and occasionally bugged entirely. Still, it's easily the most useful way to follow when the public events will come if you need to track them down quickly.
One of several ways to find a Raid group if you need Fireteam members, Destiny tracker comes with a live chatroom for putting together a team. You can pop into a chat for your console, say what mission you're looking to run, and add one another in-game. It's one of the best ways to find a Raid group if you don't have five other people to play with, and is helpful for quickly knocking out a Nightfall. Just hit the LFG (looking for group) tab and explain what you're looking for or join someone's request.
Destiny App
The official mobile app for iOS and Android is pretty helpful, providing up-to-date access to your inventory and progress, as well as the game's Grimoire. Some of these other sites make parts of it redundant, but it's a pretty sleek app that can help you out with a quick glance.