How do I efficiently use fusion cores?

The fusion core page at the Fallout Wiki has some useful information on the topic:

Actions such as holding breath when using scoped weapons, sprinting, utilizing V.A.T.S., or Jetpack modules causes the power to drain much faster, whereas walking consumes less power and remaining stationary drains none

So, don't move more than you need to, and don't move faster than you need to. Don't jump over something if you can easily walk around it. If you can just shoot something, don't bother with VATS. As @EJS has pointed out, increasing your endurance reduces the AP cost of sprinting, and this appears to affect power core consumption when sprinting as well.

This is all assuming that you're actually in the armor. To conserve power, get out of the armor when you can. When I enter an area and there's no fighting, like a town, I immediately get out of the armor until I'm ready to fast travel away. Instead of scavenging everything while in the armor, I clear an area of enemies, then get out of the armor and scavenge. When traveling long distances, you can have a human or synth companion wear the armor for you, until you get where you are going. If you really want to be miserly, you can leave the armor at the entrance to a location, fight the weak enemies, then go back and get in it when you find the boss.

In addition to these behaviors, a fusion core's charge can be extended by 10% by collecting the Repair bobblehead, and by up to 100% by investing points in the Nuclear Physicist perk.

Also, remember that nearly drained fusion cores sell for just as much as fully charged ones. So, you can sell your used ones and put the caps toward a new one. This means you should stop, get out, and swap cores when your core is low. Obviously, this is difficult in combat, so you should check your charge before engaging enemies or entering a new area. If you let a core get completely drained, it will disappear, and you will have lost those caps.

I believe that one of the simplest ways to make cores last is to just skip VATS.

I started using Power Armor permanently from about lvl 30. I have no perks that increase the duration of fusion cores. I have maxed out endurance however, which likely contributes to less fusion core consumption when sprinting. I was amazed at how little impact it actually had on my fusion cores. I started out with about 20 fusion cores at lvl 30, and by lvl 60 the number of fusion cores I posessed had actually increased. This is just from normal looting.

During this time I wore power armor 99% of the time, only walking out of it to sell loot with charisma gear.

I think that the reason for my low fusion core consumption is that I very rarely use VATS. But then again since I never use VATS I can't really say for sure if I would notice a large difference in fusion core consumption if I started using it. I just know that the way I played the game, I never had to worry about running out of fusion cores.