Just keep riding up them. I went from truly hating hills to actively seeking them out and as a result my technique and endurance improved vastly (and back to hating them again after I lost my fitness)
After a while you develop a feel for it, there's a rhythm that is very satisfying once you dial it in.
The problem with being the kind of person to push a big gear when going uphill is that it's not the kind of effort that you can keep repeating on a ride (the muscle fibres used in a big effort are different from those used when spinning, and they tire out *much* more quickly), and if you run out of gas before the top you're left struggling on a smaller gear anyway.
Unless it's a small climb that you know you can power over without losing speed, it's better to shift down early and spin a small gear at a pace you can maintain all the way through.
If you find yourself having run out of gears on a steep climb, alternate between pushing your bottom gear while sitting and shifting up a couple of gears and getting out of the saddle for a short burst. This will break up the monotony, and give some muscle fibres a chance to recover while you use others.
Finally, don't focus on how fast you are on climbs when compared to other people, there will always be people who are faster. It's rare that I find any who are slower than me but I will (usually) get up the hill.
