Your girl is just about to turn two, right? So anyone in the know is aware that this is a difficult age. My kid is basically an angel and he used to have tantrums at that age. I have a few tips for how to avoid them, but you can't avoid them entirely because it's normal for their age to test limits and to lose control of their emotions when they're frustrated, both. People may be staring because they feel bad for you, or because she's a great looking baby, or because they're tired and wish they could have a tantrum, too--you have no idea, so don't worry.
The first tip is to try to keep to the fewest number of stops on an outing as you can. You don't want to tire her out because she won't go to sleep, she'll melt down.
The second tip is, try to be at home or at a park or another place where you don't care if she tantrums or melts down, so you can let her know that when you have the time and energy, it's OK for her to have a tantrum. Some of this acting out is a scientific experiment to find out how you will react, so if you set up some conditions where you don't have a big satisfying freakout when she cries, she will lose interest. My kid had nearly all his tantrums in the grocery store in the checkout line, since that was pretty much the only place it really bugged me, and we couldn't leave. In other places, I would say, "If you need to have a tantrum now, it's OK with me," and he would cry for a minute and then change gears.
You can ask her when you go out for a walk if she can hold your hand and march across like a big girl, or you can insist on the stroller before you leave the house (good opportunity to deal with a tantrum where you don't feel embarrassed!)
Those were the days. I really miss them. I love how much fun I have with my son these days, but that toddler phase was so great.