When I pick up a DPS role, my aim is to go straight for the healers.

That being said, I also protect our healers as a priority. The same reason for both. A support can change the flow of battle. Especially Mercy or Ana. Lucio can get a few saves and save a few seconds of their team. And Zen is a pain with his discord. If they are annoying enough to take out, they're also important enough to protect. Every game has a flanker on your side, but if you're a soldier or reaper sitting by a support, you'll get into enough battles, assuming the other team is competent.

If I'm McCree, I keep an eye out for a Mercy being charged by Reinhardt so I can flashbang him, for example. Or even as Mei, I watch Roadhog carefully. If you have one that usually shoots non-stop but then you see him stop firing and not reloading, I press E and get ready to wall between him and likely a support.

You can just tell when a Roadhog is meditating and focusing to pull someone in after so long.

Of course, if I'm a tank, then it's obvious your job is to protect your team more than clearing the enemy. So that's always been a focus, especially with supports.