Since Xur turned up in the Tower last week with new The Taken King item Three of Coins, exotic farming has become a trend in the game and Bungie will almost certainly nerf the item in the future but for now, it's pretty easy to get many exotics without too much effort.
Three of Coins offers a stackable effect, meaning the more you use the better your chance of finding an exotic engram after your next ultra kill. This means you could get lucky by killing ultras on patrol, but the best way seems to replaying the base game's story missions.
Two of the best missions for this happen on Venus, 'Scourge of Winter' and 'Eye of the Gate Lord'.
Draksis – Scourge of Winter
With the former, you need to start the mission and complete the various steps in the Cinders until you reach Draksis' throne room. Once there, use a Three of Coins before taking him, and yourself, out with a rocket launcher blast. This will mean you'll restart the boss fight with any exotics unlocked remaining on the field. Repeat the process a few times and you should come away with at least one exotic.
Alternatively, kill him, jump to orbit before the timer to end the mission finishes and you'll get to fight him again and potentially unlock more exotics. This may take several attempts and if you use this method be sure to find any exotics Draksis' drops before exiting out because they will not be there when you go back.
Zydron, Gate Lord – Eye of the Gate Lord
This mission takes you to the Waking Ruins and it's pretty easy to get to the boss as you can drive past much of the quest. You need to stop off at the pillar at the beginning and deploy your Ghost, take out the hydra, and reach the boss area and that's almost all there is to it. In all, you can do this in two to three minutes at a push.
Before stepping into the ring to trigger the boss fight, use a Three of Coins and prepare to kill him. See if he's dropped any exotic engrams and if not, return to orbit before the counter reaches zero. Selecting the mission again will drop you in the boss spawn area. Use another Three of Coins, step into the ring, take him out, and see if there are any pretty yellow engrams lying around.
Three of Coins
You may need to do these missions several times to get anything and while you can use the stacking effect of Three of Coins, it really isn't necessary. The exotic engrams these bosses drop can be from Year One or Year Two.
Xur sells Three of Coins in a stack of five and they'll set you back seven Strange Coins.
Be sure to check out our guides on the fastest way to earn Legendary Marks, Infusing weapons and armor, the fastest way to raise your light level, and how the Exotic Blueprint system works. You can also find our guide on locating every Calcified Fragment in the game.