Because of marketing considerations, I think it would be best if I mark my app as free for the first, say, few weeks, and charging a price for it after that period has gone up. I think I've seen this practice on the App Store before, but I just want to make sure. Does App Store allow this? Thanks.
Yes. You can change your app from "free" to "paid" as you desire. Keep in mind users who downloaded the app for free will continue to get free updates even after you change the price (unless those come in the form of in-app purchases).