First, there is no "right" way to start. Everyone is different and faces different circumstances. Many are born into the circle, and some come come to it later in life. For those who aren't born into it, there are a few points to follow to help to make the transition a little smoother.
Before attending your first pow wow, do a little research. Review pow wow etiqutte, the arena and staff, and terminology. With these ideas firmly in place, try to find a local orginazation that holds pow wows. Good places to look are local Reservations, colleges and universites with Native American Student Associations (NASAs), or simply in the phone book. Many areas have an "Intertribal Association," such as the Inter Tribal Club of Tulsa (ICOT). Try to ask around or find people who go who can help you. Having a friend who already dances or attends is a great benefit.
After locating an organization and finding out when locals pow wows are going to be held, attend a few just with the intent to learn and meet people. make sure that it is allright that you may attend, because some pow wows, such as family dances or Formal War Dances, are not open to the general public. Don't be afraid to ask questions and introduce yourself. Most people there are nice and don't mind helping new people. Don't be overbearing, just relax and go with the flow of things. Try to observe what is going on in the arena--there is always something, like intertribals, give-aways, or Specials for recognized people.
Never copy designs or patterns of a dancer's regalia. Many of the designs are family patterns or hold spiritual meanings. What you can do, however, is observe what are acceptable designs. Learn your area. For instance, a Kiowa dancer or people who dance with the Kiowa would never incorporate an owl in a design or use owl feathers because the bird is considered a very bad omen. Most geometric beadwork patterns are acceptable, though. Learn what type of materials are used in construction. Make your outfit yours and yours alone.
Unless you plan on being a Fancy Dancer, throw out the pink and yellow feathers with those awful bent and dipped "imitation eagle feathers." If you are of quarter blood or greater Native American, look into legally obtaining eagle feathers from Parks and Wildlife, and if you are not, look to the local trader's hand--painted eagle feathers. If you choose to build an outfit and plan on dancing for a while, build it with good materials. Your local traders will be more than happy to get you what you need and may have books on outfit construction for each style. (Look here for instructionals in the future !)
Go to several pow wows before you first dance in regalia, and ask someone for advice on how to dance!
This is just a general way to get started, and you may face totally different circumstances-don't worry. The key is just to make friends, and from there more into it. Soon, the people you meet at pow wows will become an extended family to you.