Was actually working on this with the middle/high school team I coach last night. There are lots of variables in play, but here's a sort of checklist, if you will.
Approach with some speed. If nothing else, your momentum will carry you partway up with no other input. If you have a really good approach (especially a fast downhill approach) and the climb is short, momentum can do all the work for you.
At this point, you have a decision to make and in some respects it tends to depend on the specific climb. You'll have to try both to see what you like best for a given situation.
--Seated Climb-- I tend to use this one on longer steep sections and in spots where I am able to carry less momentum into the climb.
Get your chest LOW and forward to keep the front end planted and in control.
Keep your butt on the saddle, though you will probably need to slide forward so your taint is perched on the nose of the saddle.
Shift to a lower gear and spin
--Standing Climb-- This one works well when the climb is short and you want to power up it, or if there's an individual technical move in a longer climb you need extra power to clear.
Stand up on the pedals
Shift body weight forward of the saddle to keep balanced pressure on both tires. If front wheel lifts, shift forward. If rear wheel starts to break loose, shift back.
Rock the bike side-to-side to help engage your upper body in generating power
Engaging the pedals on the upstroke is not the exclusive domain of clipless pedals. You can do it on platforms (good platforms with grippy shoes), too.
As I hinted above, you might alternate between both techniques on a single climb. You might choose to use the seated climb most of the way, but stand up for shorter sections that require even more power. Every bike handles this stuff a little differently. I've got one mtb that does NOT like for me to stand up on climbs. Doing so shifts my body weight too far forward and the rear tire breaks loose EVERY TIME. After 3yrs of riding the bike, I've not found standing climbing to be good on that bike. It's kinda frustrating at times, but that's how the bike is. If I keep good weight on the rear tire, though, it's got traction for days and will claw up almost anything. My new bike works well climbing either way, which is a relief, but I've spent the past 3yrs training myself NOT to stand up on steep climbs so I now have to train myself again to stand up sometimes.
Shifting technique is going to come into play eventually. Shifting under a bunch of power is going to result in bad noises in the drivetrain and potentially broken parts if you do it too much. So you'll need to work on shifting while climbing, too. Especially the timing of your shifts. You'll want to do so right before you NEED to, when you can back off on the power just a tiny bit, soft pedal until the shift is completed, and then lay down power again. It's a subtle technique, but an important one.
