The game is rather loosely structured. As you advance through the campaign, you'll find that you can select many missions in any order you like, so long as you meet the level requirements. Your character can earn XP via playing competitive multiplayer as well, so people who dabble in that might find themselves wondering which mission to pick. You can also choose to Patrol a given planet, which places you in an entirely open world, completing randomly-generated missions you acquire from blinking communication devices scattered around. An NPC in the Tower provides bounties, which are fun side goals for you to achieve, such as killing a specific elite enemy, or earning 9000 experience without dying.
There are also Strike missions, which are Destiny's version of a dungeon instance in your standard MMO. You'll be matched up with two other players if you're not already in a Fireteam, and take on much tougher groups of enemies and a large boss or two. Personally, I loved these, as they require a little more teamwork than the average FPS, though a lot of people seem to feel like the bosses require too much damage to take down. Your opinion may vary.
Similar to a lot of modern MMOs, there are also public events that happen while you're out in the world. Most of them involve dropping a large boss or defending a point from waves of enemies, but they're great fun. Unfortunately, as of this writing they happen too infrequently. However, Bungie has said that they will increase how often they occur in the future.
There's actually a lot of co-op to be had with Destiny, but most of it will be incidental, with other players in the world who are in the same area as you. There are shades of Journey here, as you have no choice of who shows up in your game, and you can choose to hang around them or let them strike out on their own. The servers seems to cycle players in and out constantly, and it never stopped being weird to see a player disappear from your game if they moved just a little too far away from you.
This could be prevented if there were a better mechanism to invite random players to your Fireteam, but it feels awkward, and if their privacy settings are set, you won't be able to invite them at all. There's also no way to voice chat with players who aren't in your Fireteam. I suppose the game would be utter chaos if there were some kind of proximity chat, but it could make Fireteam invitations go so much more smoothly. Luckily, for Strike missions there's matchmaking, and since you'll spend a lot of time doing them later, that's a very good thing.
Destiny also supports in-game clans, but bizarrely, there's no place in the UI to show your clan roster or see their online status, necessitating out-of-game communication if they're not directly on your chosen platform's friend list.
Coming from the studio who gave us the strong, story-centric campaigns of the Halo series, it's shocking how little of it there is in Destiny. Aside from your Ghost's constant technobabble, I counted four moments in the game where it actually utilized other characters to advance the story, including the opening of the game and its ending. There is actually quite a bit of worldbuilding done for Destiny's setting, but nearly all of it is buried in the Grimoire, a series of collectible cards that are only accessible via the game's website or companion app.
In fact, if you didn't ever use Bungie.net or the Destiny Companion app, you probably wouldn't know that the Grimoire cards actually level up, and begin to provide passive bonuses to all of your characters in different ways, depending on the card type. For instance, if you level your pulse rifle card up, it grants all your characters double XP towards unlocking upgrades on their pulse rifles. Locations begin to grant twice the resources each time you harvest a node. I really enjoy the game's minimalist UI, but I can't help but feel they could have added a way to view the Grimoire in-game somehow, either via a terminal in the Tower or in a pre-login menu.
The campaign missions also leave a lot to be desired, as there is very little variety throughout. Aside from one mission which involved you picking up a legendary sword and beating the crap out of elite enemies with it, nearly every mission involved bringing your Ghost to a place in the level to scan, or bringing it to a place in the level to scan while you fight off waves of enemies. Though the areas themselves are excellent combat arenas and the scenery is gorgeous, it began to grate on me that I ended up playing Horde mode a dozen (or more) times by the end.
Luckily, the campaign is not where you'll spend most of your time in Destiny, especially once you hit higher levels. At level 18, you will unlock the Strike Playlist, which offers matchmaking to take on a random Strike mission, with extra gear and Vanguard Marks as a reward.