I'm no doctor either. But I have been an athlete of one type or another all of my life. You've been riding for less than a month and can't figure out why you're getting sore?? DOMS. You're making your body perform a repetitive movement that it is unaccustomed to. Pretty simple. If it doesn't gradually improve after 4 days of rest (depending a little on your age), seek medical attention. But its probably just DOMS.
If you want to get faster. Try intervals and wind sprints. The good Dr. surely wouldn't dare decry the benefits of intervals, especially at your fitness level (which, based on your limited fitness knowledge. I'm assuming is low. No offense intended.) BTW, you can do the intervals and sprints on your bike if you wish.