Headshots, headshots, headshots. To be successful in Halo multiplayer is to get headshots.
Move as a team when you can. Teammates can save your life and can be a great distraction.
Hold down an area as a team, this will increase your odds of survival and odds of eliminating other players.
Learn where the special weapons are. Every map has a shotgun, sniper rifle, rocket launcher, and sword somewhere. Know where these are to net yourself a positive K/D.
The minimap detects motion, while you're not moving your location won't show up for for your enemies. Equip a shotgun or sword and wait around corners for unsuspecting prey.
In maps that you have a assault rifle and pistol combination. The assault rifles equip animation is shorter than the pistol, so start shooting your enemy with the pistol and switch the assault rifle to finish them. It's even quicker to just beat them down after a few body shots.
Bounce grenades off the walls, ceiling, and ground. In Halo you'll need it to contact a hard surface before it'll explode, which also means you can throw them across the map and expect they won't explode until it touches something.
Watch your shield levels, duck and dodge behind cover as you'll want to do some damage and let your teammate finish an enemy. Most game types can be won by keeping yourself alive as long as possible.
In 4v4, keep in mind the position of your teammates and branch out into hallways to cover a larger area. Rely on their position to fish out enemies since there isn't a mini-map in swat. If a player icon goes yellow or red, they have found a potential target.
Staying together is a great tactic in Slayer, while in SWAT it is not. Staying together can make it more difficult to hit an enemy while making it easier for the enemy to get double and triple kills.
Going down sights with the DMR or pistol can be useful but in close to mid range try not going down sights for faster trigger fingers.
Keep your reticle at head level, while moving across a doorway pan to the inside and be ready to pull the trigger if you pan across an enemies head.
Learn to jump and shoot accurately, jump through doorways to get the drop on players around the corner.
Teammates can often spawn in groups, where there is one enemy there's a good chance there is more. Watch for a second enemy to run past a door or ramp.
Listen for breathing, if there are no teammates around you get ready for a fight if you hear laboring breaths. Crouch and walk to reduce your own noises.
Refrain from running or reloading more than you need to. Have your gun in ready position approaching a door or window to be able to get a shot off if needed.