Personally, I didn't feel comfortable breastfeeding in public until way into the second month.  It wasn't until I got to the point at home that breastfeeding was second nature and the baby was latching on with no fumbling did I feel comfortable going out.  Until then, I would feed the baby, then go out so I got a good hour and a half or two hours until it was time to nurse again.

I agree with the suggestion of an extra layer. My kids wouldn't stand for a blanket or anything covering them but a sweater or jacket to provide some coverage from the side worked great.  And I used to wear loose cotton shirts that would kind of hang down.  Between your clothes and the baby, it's not that hard to cover up.  And finding an out of the way bench or someplace quiet to sit helped until I got more comfortable.  I remember nursing in the car more than a few times.