Pretty much what backflip said. Each kill nets you 1 point, boss kills get 25 points and legendary bosses get 150 points. Base captures don't get you any points but do grant vantage points that you're going to want and capturing all three bases will open up the enemy core to attack.
Also, a tidbit since you said you're new, here's the grand strategy for Warzone:
It's ideal to capture the middle structure in any Warzone map, even over the initial armory run. On ARC capturing the center Garage gives you access to every legendary boss and both Soldier Commanders and two Knights that spawn there later (550 points worth of bosses). Capturing the Spire on Apex gives you premier access to all three legendary bosses, the Hunter Elders, the Knight Luminary and the Banshee Raiders (575 points worth of bosses). Capturing the Fortress on Stormbreak gets you access to Commander Lochagos, Captain Wamik and the Wraith Commander (325 points worth of bosses) and capturing the Monument on Noctus will get you all three legendary bosses, both Hydra Soldiers, four Hunters and two Phaetons (650 points worth of bosses).
Furthermore, on many of the maps you can completely and utterly restrict movement and out slay the opponent with your superior vantage point. On Stormbreak if you control the Fortress and one armory you have unrestricted access to every part of the map and can completely prevent the enemy team from moving. The Monument is an important enough vantage point that the enemy team will be rushing all game to capture it, and a successful repelling of boarders will still net you points.
No one can argue that they help, though that's mostly due to the size of the maps themselves. It's not difficult to manufacture situations in close quarters where you have the upper hand. Longshot AR is still pretty bad, though. The accuracy of the rifle isn't increased, just the magnification on the zoom, so you're still spraying at a dude that's like 50 meters out. The DMR is ideal for most infantry engagements and the H2BR comes in a close second.
It's important to note that that's if all else is equal. If you're not good with the BR you're not going to be better with the DMR. The BR is sufficient for pretty much every possible engagement in Warzone, I just prefer the extra range afforded by the DMR.
i'll give my students a holiday in honor of your troubles.
