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I had shin splints when I started running. I stopped and started so many times due to the pain in my shins. I finally got these compression sleeves from McDavid (they have rubber tubing that compresses the area on the shin where it hurts) and they eliminated the pain. After 6 months or so, I didn't need to wear them anymore.
As an athletic training student, I've learned that stretching the fronts and backsides of your lower legs with a foam roller, and getting some arch support in your shoe if you're flat footed or an overpronator will also help.
As far as the burning and fatigue go, that sounds like normal beginner runner stuff. The more you run, the better your body will oxygenate your leg muscles (by growing more capillaries and using the muscles more efficiently) and I suspect those issues will disappear.