My 13 year old DD is a figure skater. She started taking group lessons (called "Learn to Skate") at one of our local rinks when she was 7 years old. The classes run in 6-8 week sessions (varies), then the coach gives them an observation skating test and they pass up to the next level. After about 6 months of that, I approached DD's favorite coach and asked about private lessons. We've been doing private lessons ever since with that coach and her twin sister. Sometimes DD goes skating with her non-skating friends or attends birthday parties at the rink and when her friends see her spin and do double jumps, they all scream with excitement.
Skating is a great sport. DD started doing competitions right away. Skaters can compete even at the most very basic level. We just got back from a huge competition last week. Aside from competitions, there's also skating shows, exhibitions and test sessions to participate in if she doesn't want to compete. She loves getting pretty skating dresses that we bling up with lots of crystals. I spent a lot less on them when she was a beginner and learned to buy used ones on ebay, but now we spend around $200 per dress on average. It is a pricey sport though when you factor in ice time, lessons, skates, dresses, and competition fees. It wouldn't be too much to get her started in group lessons though just to see if she likes it. Here's what it would cost to get going:
Beginner figure skates: $100ish (or she could use the rink rental skates at first to see if she likes it. If she does like the skating, I'd get her out of those skates ASAP. She'll love the sport a lot more if she's in good skates).
skating tights: $18.00 (or she could wear stretch pants and just wear some nylon socks, but the skating tights help keep her legs warmer)
practice skating dress (optional): $ $30 - 50 or less. They are all over ebay.
group lessons -- $85 per 6-8 week sessions 2X's week (some rinks also throw in a free public session each day of the lesson for extra practice time)
skate guards & soakers: $30 (I can help you out here if need be)
A bag to carry her stuff in. Most skaters use Zuca bags
http://www.zuca.com/ but she wouldn't have to have this. I found one on Craig's list (new) for $20 when my DD's started getting a lot of wear & tear on hers. Some girls use small carry-on roller suitcases, too.
If she really likes the sport and moves into private lessons, then those would cost anywhere from $15 - $25 per 20 or 30 minute lesson, depending on the coach. There are coaches who charge two and three times this amount if you want an olympic type of trainer, but for a beginning level, you should be able to find one for less. I'd ask around at the rink, and observe during the group lessons.
Most beginners start out just taking one or two private lessons a week. I'm a board member of our skating club, so if you want any more info on the sport, feel free to PM me.