Leveling fast in Destiny is something that every player wants to do and right now there is a lot of incentive to as players want to switch between the different classes without a lot of trouble. Since you have to level each one up yourself, it can become sort of mundane, but never fear with the following tips you should be able to hit max level without having to experience the entire game all over again.
I suggest for anyone just starting to avoid this guide in a way - Destiny is great and most of the fun you'll have is learning and exploring and following the breadcrumbs. However, if that isn't for you, keep on reading.
It's important to understand the very start of the game, until the Crucible unlocks and you get a few of your starting skills, to just follow along with the story. It's suggested that the first four or so levels just be done via the story. This takes 30 minutes or so of your time and gets you some starting gear, unlocks all of the features that you'll want, and gets you ready.
The Easy Way
Just do story missions. They're one of the fastest ways to level, but it can be very boring if you've gone through the process already.
Bounty Farming - The Fast Way
Once you start getting bounties there is a simple process to follow from here. Get all of the bounties that you can, examine their objectives, ditch the ones that are going to take too much time (farm an unnecessarily large amount of materials or whatever), and complete them.
There are some rules to follow. Never do a mission above your level, this is going to take way too long and not be fruitful. You're going to have to unlock the other planets, this generally involves completing the mandatory quests/missions in order to do this. Each bounty that you do is going to give you a huge chunk of XP/rewards, in addition to the mission reward.
Some people say as soon as a bounty is done, abandon the mission since you'll restart at the checkpoint and this could be beneficial at a later point. Some say finish the mission. I'm indifferent, I generally finish the mission since I'm already there, but if it's either a really low level mission or a really difficult one, I'll bail once the bounty is done.
You do not want to do anything that is going to just make you stay stuck and bounty refreshes are going to be your only hindrance, but it's always assumed you'll take a break to let bounties restock plus you'll get bounties in various missions.
If PvP is more your thing, you can do the bonus XP playlist in Crucible then do the story missions / bounties at a later point. This only works if you're good at completing Crucible bounties, of course.
If playtime is short and you want to level up fast, but over multiple days, just login and do the Cosmodrone daily public quest, bounties, and any easy to do missions. Just do this each day as time permits and you'll level up rather quickly after a few days. Perfect if you have a main and just want to level something up on the side.
Tips & Tricks
When starting out, don't bother killing enemies. Your level is automatically adjusted. For the first mission, just walk literally to the end, because nothing on the way is going to prevent you from moving forward and you'll automatically reach level two. This will save you a huge amount of time.
Fighting enemies is great XP, so don't skip them, but also don't get bogged down in high traffic areas with high respawn rates fighting enemies that aren't preventing you from moving forward for the mission. Unless you want to, of course, but if it's not a bounty or a mission, you don't technically need to bother.
Bounties are great - they even give your weapons XP when you turn them in, so make sure to have the right subclass/weapon/armor equipped that needs XP when you turn it in.
There isn't a way to really "exploit" the game to get to leveling faster. It's more about being smart about missions and bounties. If you've played through the game before, you really sort of know which missions are viable for you to finish and which ones you'll only want to load up to get a bounty over with. In addition, you'll want to