I nursed my son till he was 2.5, and I nursed him EVERYWHERE. I was very discreet while doing it - the more you do it, the better you get at being discreet and effective.
There are nursing covers that do a great job. When the baby is that small, it's kind of easier because they don't play with the shirt, etc. while you nurse. Get a nursing camisol - it covers your tummy like a regular camisol so you don't feel naked, but the breast part is un-hookable. They're great, especially in the winter. I really felt that they helped because they covered the maximum amount of skin, gave access for the baby, kept me warm, etc. Unless you're pulling your breast out of the top of your actual shirt, like pulling your shirt UNDER your breast, the baby's face covers 95% of your chest anyway. In order for anyone to see anything, they'd have to be all up in your business - and then you give them the stink-eye (I honestly don't remember this happening except for VERY small children, and they're curious - you just tell them you're feeding the baby and need privacy).
Find a comfortable place where you can have a bit of privacy, or park as close to where your other kids are playing and sit in your car with the door open so you can watch your child. I would just sit on the bench and nurse. Most people at parks are moms playing with kids and either won't notice or care. Few are dads, and they would act the same, generally.
For restaurants, I'd pick a booth in the quieter spot, and sit on the inside for extra privacy. Stores - nurse in the changing rooms (did that a lot at target).
ADD: Funny nursing story. My theater teacher from college had a baby after I graduated. I visited the school for a homecoming event and got to talk to her. She told a story about breastfeeding. Her baby came to class with her, as her office was connected to the classroom. She was teaching while holding the baby, and a blanket was covering it. She was NURSING while TEACHING and the college kids didn't even realize it.
I think I realized then (15 years before having a kid) that you can nurse pretty much anywhere with a little creativity, and it doesn't have to bother anybody.