You have a great advantage (i know it's not gonna make you feel better) in terms of being able to throw weight around. As you pedal, try rolling your shoulders to your pedal stroke and shifting your weight left and right.
It'll help since you're using not only muscles but also your body weight (which you have a lot of) and momentum to push down on the pedal with each stroke.
Also work on the breathing, as someone else said, exhale fast and inhale deeply. Try to time it so that your breathing is in sync with your pedaling; it helps me get into the groove. Don't look straight up, instead, look forward maybe 10-15 feet and look up occasionally to check for stuff in the road on a long climb; it's easier on you mentally. Also, knowing when the hill will end works too since you know exactly how much farther you have to go.
Try counting numbers. Count backwards from 10 each time you put your foot down, repeat when you reach 0. It's a mental game but it splits the climb up into little 10 stroke increments.
Spinning works the aerobic system more. Mashing at a low RPM works your muscles more. Since you really haven't put that many miles in, I suspect neither of them are too strong right now. Spinning's easier on your knees so you should work on that. Try to keep things above 75rpm or so (if you have a cadence meter).
On long climbs, don't pay attention to speed, try to find a gear where you are right at the edge but never going over it, where your legs aren't like giving up after every pedal stroke. Get into a rythm that you think you can maintain for a while and stay there. With the correct combination of gears, you can get up almost any hill, just a lot slower, but it's possible. Your body's capable of a lot more than you think it is, half the challenge is mental and not giving up.