Our experiences is only with dance so I'll address that with some tips:
Look for programs with names like "creative movement" or "tiny tutus" etc. -- they tend to be dance for the very youngest. Be sure to go observe a class without your daughter there, before you sign her up, to see if the teacher has a style you think would mesh well with your child. Do not hesitate to ask questions of the teacher afterward, like how he or she handles a child who wants to go out to Mom, or how they handle kids who get frustrated. Anyone who teaches movement or dance to young kids should not be fazed by questions.
(And by the way, I agree that Irish step seems too precise for a child this young. You may find some class offered for her age but definitely check it out first and think of her personality -- would she enjoy the precision of that, and follow the directions, or would she prefer the freer movement of creative movement or tiny tutus ballet? You want her to succeed and have fun, not feel she's "getting the steps wrong," at this age, or she'll be put off all dance.)
Don't look only at dance studios; many community centers, county recreation programs, etc., offer dance classes for kids of this age and those are usually short enough -- six to 10 weeks or so -- that your daughter can try it out and it's not a huge commitment. If she loves it, you sign her up for the next 10-week session or whatever. We did many, many 10-week sessions at the recreation center until my daughter was old enough for a dance studio and she's still dancing at age 10.
Same goes for gymnastics-- enroll for a six-or 10 week class if you can. For either activity, you don't want to enroll her for endless weeks of classes because she's only three and you want her to try new things. If you have time during the week, there's no reason she can't do both. At her age, neither should have more than one class session per week anyway, so why not try them both? Lots of exercise!
As for gymnastics versus dance, no need to make her choose quite so young. However, if she likes music, dance can do a lot to help children appreciate music even more. They really learn to listen well to music if they continue in dance, because they must move along with the music.