How old should a child be before we start disciplining them and what are some ways to do that?

I have a 2 year old who acts like a regular 2 year old-----saying "no" to everything-----wanting to do things her way, running away when I want her close, throwing food on the floor, etc. Do I just let her grow out of the "terrible 2's", or do I attempt to change that behavior?

I have a 3 year old daughter who is a terror to take places. I'm embarrassed to take her places where she needs to sit still. The only way I can get her to obey is to give her a new little toy, etc. This is exhausting and also limits where we can go. Any suggestions?

My sister and I were hit by my Dad when we were little kids and his temper was flaring. I now have the belief that hitting a child is bad. But then how do you discipline a 3 year old who runs out into the street? Is it ever OK to hit (or I guess spank) a child?