If you teach yourself how to skate, don't expect Dorothy Hamill-type perfection right off the bat. Control your movements with small steps, holding your hands out and leaning slightly forward. Children may have an easier time of it if they push a folding metal chair while they skate. Learning comes faster and easier with a quality pair of skates. Bargain skates are available, but often fail to provide adequate ankle support and can lead to injury.
Figure skates are easier to balance on than hockey skates and are therefore better for beginners.
Another key to proper skating is a good fit. Your skates should generally be one size smaller than your street shoes. If you can't walk in your skates without wobbling, try a smaller pair. Many rinks rent skates, which provides an inexpensive means of testing for the proper size. For children, renting is an especially good idea since they tend to outgrow their skates rather quickly.