There are a handful of basic concepts to remember when playing Vanquish:
Do not let your meter bottom-out (avoid going into over-heat as much as possible). Basically, learn to keep an eye on the circular pie-graph near Sam's body when you trigger AR and always drop out of AR right before it ends. The meter refills fairly quickly when it's just red turning to green, but an overheat causes your AR to be completely unavailable for about 8 seconds. Use lots of tumbling/ukemi for movement while waiting for the meter to recharge.
Try to avoid using the cover mechanic, instead, rely on your tumbling/ukemi and the ability to trigger AR straight out of a roll. During a run, the game consistently tells me that I use cover about 3% of the time. Cover can be abused though as you can use the snap-to function to close gaps and then that can be chained into a vault where you can then trigger AR while in the air.
Avoid using melee (it instantly sends you into overheat) unless you're using the saw-blade weapon… in which case, you can throw infinite punches. I will occasionally use the run-up-vault-kick when a boss-type enemy is low on health to speed up the kill assuming there are no other enemies in the immediate vicinity.
During a run, I only use 3 weapons (well, 4 if you include the required usage of the sniper rifle during the one stealth auto-scroller)“ The assault rifle, the heavy machine gun and the rocket launcher. This is important as all of my weapons level up very quickly since I'm not constantly swapping things out. So try to find a handful of weapons that you're really comfortable with. The assault rifle is fairly accurate at long distance and has loads of ammo. It's great for taking care of the basic red AIs and works well for headshots. The heavy machine gun has a lot of stopping power and is really good for dispatching larger enemies (like the Romanovs) as well as boss weak points (when fully powered up, it can decimate a KNRB's limbs and core). The rocket launcher has massive impact damage combined with useful splash damage.
Reload cancelling is another important tactic. Either reload during a tumble/ukemi or if Sam goes into the reload animation for whichever gun you're using, quickly switch to another weapon and then switch back (don't use this tactic with the rocket launcher though).