How do you handle ongoing effects or damage, especially if a PC is near death and is hoping he can get healed or have an ally dispel the effect before his next turn comes up? Does the PC just pick a slower action and hope for a low initiative roll?
It's pretty simple. All turns happen concurrently within the round, so any rule which says "at the beginning of your turn" happens at the beginning of the round (before actions are declared), and anything which happens "at the end of your turn" happens after all actions are resolved. So if you knock someone unconscious on round 1, they'll make a death save at the beginning of round 2, and then you'll declare actions for round 2.
This does mean that certain ongoing effects like a monk's Stunning Strike get better: a monk who rolls a 28 on initiative and then manages to stun a vampire will
both prevent the vampire from acting on round 1 (because he's stunned) and give advantage to all his allies on round 1, and then on round 2 the vampire will
still be stunned and unable to act. So high initiative matters a lot, as well as
[tangent] high intelligence (because characters declare actions in order of Intelligence, lowest first, highest last). And I'm totally okay with both those effects.
Interestingly, I've seen lots of players pump Intelligence even if it buys them no mechanical advantage whatsoever. E.g. I've seen Death Clerics drop an ASI into +2 Intelligence, and a Shadow Monk/Druid boosted his Int from 9 to 12 over the course of two ASIs, after he'd finished maxing out his AC. I asked the players about it and they confirmed that they really like being smarter than the monsters as much as possible, so they can plan their actions more effectively and avoid being surprised.
[/tangent]
So anyway, there would be no point in the PC picking a "slower action" in order to get healed first. What he is really hoping for is that his buddy will cast a healing spell or Otiluke's Resilient Sphere on him before the
bad guy's initiative. His own initiative is pretty much irrelevant unless he's trying to e.g. Disengage before the bad guy can hit him.
P.S. The way I implement Delay is that you can declare "Delay" as your action, which means you automatically lose initiative to everyone who is declaring now, but you get to declare your action after everyone else's actions are resolved. You're basically waiting to see what happens before committing to anything.