Hill training and hill repeats are a great way to get into superb running shape and improve your VO2 Max (or the amount of oxygen your muscles use during cardiovascular activity; the higher your V02 Max, the more oxygen you use, which equates to the more energy your body can produce, translating into better or increased aerobic shape). Elite and professional runners all train on hills to improve their performance. So in short, yes, running hills is definitely a good thing to incorporate into your regular training schedule! And if you are worried about getting injured, just be smart. Running up and down hills does create some unusual tension on your leg muscles that you wouldn't otherwise experience so be smart about strength training those legs. And maintain proper form as much as possible - head and vision at a 90 degree angle, pick up your feet as much as possible (no shuffling!) and PUMP THOSE ARMS! The faster you pump those arms, the quicker your leg turn over will be. Plus, it gives you something to focus on mentally.
In fact, you are actually more likely to injure yourself on a downhill gradient than you are on an uphill one. This is because your body is trying to control itself more from toppling over, while not slowing down too much. It often means your quads and knees will take a beating because they'll shoulder much of the impact. Again, strength train and focus on those quads and the surrounding muscles to make sure you're strong enough to control yourself downhill.
The best way to increase your running speed through hill workouts is by short, fast sprints up a hill. It doesn't require sprinting longer than 10 to 20 seconds at a time. But sprinting as hard as you can on an uphill gradient will really get your heart rate pumping - and drastically improve your V02 Max! Like we said earlier, an improved V02 Max means you'll be in better shape, and you will be able to pump your arms and legs that much faster on a flat surface without losing speed or getting out of breath. This translates to faster race times!