As alpinerabbit said, try to keep your cadence somewhat consistent. Cadence being how fast your legs are spinning. When you first feel it getting harder to spin at the same cadence, start shifting. The more you ride, the more you'll anticipate when it's time to shift, but it's more by feel. A good cyclist shifts often as the road changes. If you are riding with a group, you'll often hear people shifting right about the same time.

As for how to get up the hills faster, this is something I have worked very hard at for 3 years. I still feel like a slow poke compared to the group of people I ride with most of the time, but I have improved immensely. Here are some points to ponder:

1 - You'll get better with time and experience on the bike.

2- Relax your breathing and sit in a position that doesn't compress your lungs. (Study up on proper sitting position.)

3- Some people find standing helps, particularly on a short climb or at the top of a longer climb. Standing makes you go faster, but it also takes more oxygen, so you start breathing harder faster, so pace yourself with the climbing. Standing on a climb doesn't have to be all out pushing it, though. You can stand while in a harder gear than you were in while sitting and try to "take it easy" to some degree. I often stand for short bits on a long climb to relieve my aching muscles (standing uses different muscles) and/or to relieve the hoo ha! Get blood flowing ... wherever it may need to flow.

4 - Make sure you are pedaling efficiently, giving power to the pedal throughout the entire revolution, not just up and down, but across the top and bottom of the stroke. Should be an even circular flow.

5 - Proper bike fit can make a big difference in performance.

6 - Don't grind. A lot of beginner riders pedal really slowly in a harder gear on the hills. This wears out your quads faster. Spinning faster in an easier gear takes practice and works your lungs a little harder, but gives you longer endurance and better strength to climb in the long run.

7- However ... having said that, purposefully riding in a hard gear for "training" purposes, can help build muscle and improve climbing over all. I just talked to a coach last night that suggested going up a long 2% grade (just a little bit of incline), starting out in an easy gear and switching to the next harder gear every 2 minutes until you get to the hardest, which might mean you are pedaling and moving REALLY slow, and then work your way back to the easier gears.

8 - Intervals. Warm up for 10-15 minutes, then find a hill that is not too long, maybe 1/4 mile or less, and push your limits on it. Ride it as hard as you can, then recover, go to the bottom of the hill and do it again. You might stand on the hill one time, sit and ride in a hard gear, spin fast in an easy gear, or just ride a cadence that feels good, but repeat the same hill at a high intensity and mix up what you do. Be sure to recover between each hard effort. You may only be able to do this a couple of times before you feel completely wiped out, or you may do it 5 or more times.

9 - Fill hilly rides and make yourself do them. I rode Heartbreak Hundred this year, a hundred mile ride with 8,500 feet of climbing (in case you don't know, that's a TON and I never thought in a million years I'd ever be insane enough to such a thing. But dh insisted I "could" do it and I "made" myself do it in an effort to get better at climbing. I wasn't very fast, and it got pretty miserable for a while, but I accomplished it.

10 - Develop a positive attitude about the hills. They aren't going away. They can be downright hard to climb, but getting frustrated about it doesn't help. When I learned to accept the hill and stop griping about them, even though I was still slow, at least I was less frustrated! In fact, on one ride where it seemed like there was one hill after another and I was SO done after riding 65 miles so far, I started getting goofy, saying things like, "I LOVE hills! Hills are my favorite! Bring on the hills! That's MY hill!"

When I look back at my stats from last year to this year, it's amazing how much I have improved, partly by working at it, and partly by just getting out there and riding a lot. I ride with a friend who is good on the hills. Sometimes I try to stay with her, especially on a shorter climb, and it helps me push myself. Riding with people who are a little better than you can greatly improve your riding. Watch what they do. Shift when they shift. Push yourself to keep up when they start to get away from you (within reason ... you have to learn to pace yourself). Sometimes for me it's just mental. I CAN go faster if I TRY hard enough and have a "carrot" dangling in front of me. I call my friend my carrot. I get on her wheel and just try to stay there as long as I can. When I ride by myself or she gets way ahead of me, I unconsciously go slower than I should sometimes. So focusing on someone else's wheel that is just a little faster than I typically go helps me.

You can do it! Give it time.