Boss takedowns and base captures will net you the most points quickly. I am 85% sure that base defense kills (kills inside an armory/garage/etc) will net you more points than regular kills. Any time your team get a boss kill or caps a base, you get a smaller portion of those points, but you seem to get way more for being present.
On Stormbreak, rush Lochagos with a Suppressor, Boltshot, and/or Splinter grenades. They’ll tear him down very quickly. A Covenant boss spawns on the loading docks near the armories early on, and Knight Luminaries in the ice caverns. Those are all good early bosses for racking up points.
On ARC, try to get to the Sangheili General and assassinate him early. He spawns less than a minute in, I think. Then, cap your armory ASAP, and go after the Ranger Commander before rushing the garage. Do it right and you can get to the garage after all of the AI are wiped and you can focus on killing enemy spartans. There are some promethean bosses that spawn around the Crane Pit shortly after that, and both are fairly open access, so be aware of enemy players going after them. Let them do some work if you can and take them out first.
Apex 7 is littler trickier, because people push hard for the Spire, but Hunter Elders have tons of health and your REQ levels are typically pretty low that early. Go after the Zealot assassins in the caves first and grab their swords. Spartan Charging a hunter with a sword will stun it long enough to boost out of melee range and double back for another hit and the swords do significant damage. This is still risky, and shotguns may work better if you can get REQ level 3 earlier on. YMMV.
Noctus I’m still learning, but the first boss just outside of your home base can be attacked from behind. Hit the man cannon on your way out and ground pound the stairs across the main road. You will likely destroy the door, so be careful of the elite turning around on you if your allies aren’t down there yet. Take him out, and go straight to the Hydra Soldier spawn on the outer arms of the Monument, half way to either garage. Then go and take your garage.
Using these methods (and continuing to track down boss kills), I have been finishing about 2/3rds of my recent games with over 1000 points. It’s not 100% effective, because you’re competing for these things, but you will develop a routine and get better at it. I’m still working on my timers.