You’ll get the exotic engrams as random drops. Maybe one exotic per 10-20 legendaries?
In addition to the blue flag on Earth mentioned by divedivedive, there’s a exotic weapon quest associated with the blue flag on Nessus. Both of these are kind of annoying, and both give an legendary energy weapon and exotic kinetic weapon. So the efficient way to play is to advance your other equipment by e.g. milestones between doing these two quests.
Then there’s the Rat King which I think starts from the Titan blue flag. But the steps you need to do after finishing the actual quest are literally impossible to finish by yourself. They need to be done in a team with other players who are doing or have done the same quest.
Subsequent playthroughs will be faster since you can share weapons from the other characters. (They’ll also be worse, since the flaws of the campaign are more obvious the second time through). I got my third character to 278 in a single 12 hour session, and still have to do any of the exotics or powerful gear quests.
Was going to do them just now before start a raid and finish at about 290. Except we’ve been hit with multiple hours of maintenance at the most inconvenient time possible for Europeans, third week in a row :(