Admittedly, when my first child was born nearly five years ago, I felt apprehensive about nursing in public places. I started going out with my baby immediately after her birth, but the first few times I rushed back home as soon as my baby seemed to be getting hungry. Then, when she was around five days old, I was waiting in the pediatrician's office and she was hungry. If you can't nurse at the doctor's, then where can you? So, out came the breast. I've never looked back and have breastfed everywhere. Personally, I never covered up.
Breastfeeding in a public space should not be any more controversial than bottle feeding in public (although it is). Covering up is annoying for mom and baby, and hiding away in a bathroom is even worse. Would you like to eat in a public restroom? Not me! I only received a couple of negative comments, and a few more "shocked" looks. But actually, breastfeeding in public has been pretty smooth for us. Breast milk is normal food for babies, and breasts are, after all, there for breastfeeding. If you do want to cover up, there are several good methods.
You can use one of those special nursing covers. You can wear a tank top underneath a t-shirt and lift the t-shirt up while pulling the tank top down, and then drape the t-shirt over the visible part of the breast. Most people will think your baby is sleeping. Or, if you use baby carriers like a Baby Bjorn or Ergo baby carrier, the carrier will cover your breasts when you use the front carry and nobody will see. You might also like to read about early pregnancy signs while breastfeeding.