No mountain (where there is a road) is to steep to cycle. If it seems to be, you're either a) nearly dead, b) shifting too late, or c) your bicycle is geared to high.
I have a 2006 or 2007 Giant Cypress, and it has 24 gears, so that's 3 on the front and 8 on the back. I will admit that sometimes going up a hill, I am out of juice before I am even in the lowest gear, but I cannot change to the lower gear because I just cannot pedal another stroke.
Now, here we have it...
Aim for a cadence of at least 80 rpm. If you cannot maintain that, shift down. If you're in the lowest gear and still can't maintain that speed, get out the saddle!
Edit:
A lot of beginners have their saddle too low. This will severely impair the amount of force you can transmit, and it will greatly enhance fatigue. Make sure your saddle is at the correct height.
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What's this about clipless pedals? I don't see any clips on my pedals. Tell me more...
Well, in the old times, your biking shoes would have a kind of leather wedge on your sole, that would grip the metal bars of the pedals. Pedals would also have a hook ( the "clip") in front, that held and positioned the foot. You would then strap tight a leather belt, firmly securing your foot on the pedal.
With your feet firmly fastened to the pedal, you could apply force not only when pushing the pedal downwards, but you could also actively pull the pedal upwards during the other half of the pedal cycle. Transitions beetween the phases of the pedal cycle are also much smoother. And you could apply absurd force on the downward phase of the stroke, as you need not fear to slip off the pedal.
On the other hand, before coming to a standstill or getting of the bike, you would have to bend down while riding and loosen the leather belt - otherwise you would not be able to get your feet off the pedal!
Then came technological progress. Enter the clipless pedal.
These "clipless" pedals have a mechanism that locks to a wedge / potruding thing on the underside of your cycling shoes. Twisting your foot sideways unlocks it. This takes some practice (whenever I restart cycling after a longer pause I set myself against a lamp post and practice unlocking over and over, else I'll fall on the first stop...) - still it's much safer than the old fiddling with leather belts.
These come in different styles, some with bulky wedges (look system), some smaller. For everyday cycling, I'd suggest the Shimano system, as ther mechanics are small enough to fit inside the sole of a shoe - so with a mountain bike shoe, you can walk rather normally (cycling shoes have stiff soles, so walking is rather uncomfortable, but definitely better than with road racing shoes).
However, in the stage you're in, I would not recommend them yet. You still have to learn shifting, you seem to have to learn how to be comfortable on the bike, if you add "holy, shit, I can't get my foot of the pedal at the traffic light *PANIC!!!!*" now, that would simply be too much.
You could, however, add old-fashioned clips
without leather straps. They would guid your feet, making sure you rfeet are positioned in the right space - something a lot of new riders make wrong. (You want the pedal to be under the balls of the foot, not under your middle foot).
example:
http://www.bike-mailorder.de/Bike-Teile/Pedale/Pedal-Haken/MKS-Half-Clip-Pedalhaken-silber.html