The best way to get exotics is to maximize doing activities which have a higher chance to drop them. Do these things for maximum chance:
- Do all Powerful Rewards on all characters. Every single Powerful reward has a chance to be an exotic instead of the normal reward. So do all your Crucible/Strikes/Gambit/Heroic Story/Heroic Adventure/Flashpoint/etc... - each day that one of these resets, do it again
- Do all Dreaming City activities every week (Ascendant Challenge, Blind Well/Offering to Queen bounties, featured story mission, bounty for 8 daily bounties).
- On Curse Week, do Shattered Throne on all characters
- If you Raid, do the Raid every week on all characters
Once you exhaust all your powerful rewards (I'm sure there are some I forgot to mention), then you are going to be limited to hoping one drops in the Wild as an engram. Focus on activities that have a lot of enemies - the more enemies you kill, the more chance you might see one drop.
Just doing all my powerful rewards this week, I got Trinity Ghoul, Ursa Furiosa, Shards of Galnor, Geomag Stabilizers and Queenbreaker (my luck this week is not typical but if I had not farmed all my powerful rewards, I would have never gotten them)
The easiest way (but probably the most time consuming) is to buy a bunch of vanguard boons from Zavala and use one at the beginning of a strike. Need 2 people in your fireteam. Quit out and repeat until you get your exotic reward pop up.