The third law of ice skating is "Keep your hands out to your sides" (but don't wave them about). That may seem like a strange rule, especially if you see advanced skaters running around with their hands in their pockets, but there's a lot of sense to it.
When you are just getting your confidence on the ice, and you are feeling like you might fall at any moment, keeping your hands out to your sides helps give you better balance. Like a tight rope walker using her arms to give her extra balance as she crosses from one platform to the next, you can feel more confident and in better control with your hands out.
On the other hand, you don't want to wave your arms about unnecessarily while you skate, as that will do exactly the opposite. There's a little demonstration I do for new skaters where I move forward by using just my feet, then I go back and, with my feet firmly held together, I wave my arms around and I move forward again. Then I show how moving your feet and arms in opposite directions, as you normally do when you walk, actually sets you in opposite directions top-to-bottom, and makes you more likely to fall.
The thing about ice skating, and you'll hear this from me lots, is that it's all about the small, subtle things that you do. When you make big movements or add extra things into the equation, you are more likely to fall than to do what you wanted to do.
And speaking of small, subtle things you do on ice, having your hands out helps you to steer your body better. Whenever you want to go around a curve, you can put the hand on the outside of the curve in front of your body and the other hand behind you, and you will go around the curve much more efficiently. If you watch figure skaters, you'll notice that they do that while they are doing crossovers on a curve. Hockey skaters will have their hands on their sticks, but if you pay attention, you'll see that they are actually doing the same move with their shoulders. I call this "hugging your circle".
In life you have to keep your balance, too. Know how to move forward with a minimum of effort and maximum results. Another way to say that is, "Work smart not hard." When you work too hard on something that could have been done more simply, you wear yourself out and you might not accomplish what you need to at all!
If you don't know how to work smarter? Ask your friends who seem to be doing it better than you are, or get yourself a coach of whatever flavor makes the most sense for your situation. Remember, coaches do two great things for you: they teach you how to do things more effectively and they push you to do the things that you want to do but don't think that you can. You can! So get out there and do it!!