As a dance teacher myself, I am a little biased, but if she is interested in wearing a tutu and ballet shoes, let her try it! Age 3 is the perfect age to start looking into extracurricular activities for your child. They usually have enough coordination to enjoy most things like gymnastics, dance, etc...
If you just want her to get a taste of it for the first time, maybe look at your local recreation department for ballet classes for preschoolers. Most of these are 6-8 week courses, and don't require costumes, so you won't have those costs to worry about.
If you take the studio route, do some research first and find one that maybe you can take by semester instead of committing to a full year. That way if she doesn't enjoy it, you're not invested in it until spring. Different studios have different rates/prices, so shop around and get references from other parents. Also ask about their recitals - how much pressure is placed on the 3's to "know" their steps. Some studios are very strict about this (which in my opinion is ridiculous for preschoolers), while others are not, and just want the children to have the thrill of wearing a costume and dancing on "the big stage". I would look for a studio who wants to teach their 3's how to move, work as a group, and follow instructions. This is the foundation of a dancer and 3 year olds can grasp this concept if it is done in a fun and loving manner.
Also, many studios offer creative movement classes for preschool age children which is a taste of dance, but focusing more on coordination and development of skills such as hopping, skipping, etc...
I personally enrolled my daughter at age 3 in a ballet/tap combo class. She loved it (which made me really happy), but the day she decides dance isn't fun anymore, that's fine with me. Let her try a few things to find out what sparks her interest! Good luck and enjoy her while she's little! They grow out of tutu's TOO fast!
:)paige