The only surefire way to obtain exotic gear is from Xur, Agent of the Nine. This NPC appears in the tower from Friday to Sunday 9:00am PST. He will sell you exotic gear for Strange coins, and an engram for Mote's of Lite.
The armour (such as helmets, gauntlets etc) will cost you 13 strange coins. These are sold as ready to use pieces.
The exotic weapon (which you'll need to upgrade for the 'Packing Heat' achievement) will cost 23 strange coins.
You can also buy an Exotic Armour engram for 23 Mote's of Light. I have heard reports that this engram always makes exotic gear, but is not guaranteed to be the same class (I.e. hunter, titan, warlock) as your character.
If buying this engram, buy it at your own risk.The best way to earn Strange coins is by doing weekly Heroic Strikes. They are as follows:Lvl 22 Strike earns 3 coins,
Lvl 26 Strike earns 6 coins.
Lvl 28 Strike earns 9 coins.
(That's 9 total. You can do each for 3 a piece or do the lvl 28 strike for all 9 at once).
You can also gain coins by chance by:Decrypting rare/legendary engrams with the Cryptarch.
Looting chests.
Playing crucible matches.
Playing a one Public event a day and earn a Gold rank.
(This is also do-able across multiple characters each day, so if you have more than one character do one event a day with each one).
Mote's of Light can be earned by leveling up after you reach lvl 20, 1 for each lvl up.
Again there are also possible mote's found by chance as follows:Decrypting rare/legendary engrams with the Cryptarch.
Looting chests.
Playing crucible matches.
Daily Public Events (Gold tier rewards).
Killing Major & Ultra (Yellow Health Bar) enemies.
Hope this is useful guys!
EDIT 1: Another way to earn exotic weapons/armour is by finding exotic bounties. The specifics of obtaining these bounties is still a bit murky, but some people have reported finding them after doing 25 strike missions.
* Cheers to xTGE for the info
Some of the bounties you can select and how to accomplish their goals are as follows in these links:
http://gameguidecentral.com/destiny-exotic-bounty-guides/http://destinynews.net/guides-exotic-bounties/DISCLAIMER: Weapons, armour and other items are subject to change with each new appearance of Xur, Agent of the Nine. Please remember the value of cost/items available may be subject to change due to the nature of this game/NPC.Also; for those having trouble unlocking glitchy achievement do this:
Press the Menu button RB over to settings, highlight Log Out and choose Switch Profile (not Change Character). This will take you back to the game start screen, and once you press A, it will ask you to choose your Xbox Live profile.
Choose your profile and then choose your character from the character select screen. Any achievements that should have been unlocked will then unlock.
*Cheers to FLEEBS & ludo10109 for the info