the hardest part of being a parent is teaching our kids to be independent. She's 3, and it sounds like she's always been with you. If so, this sudden change is going to be a struggle for her, but the fact that she's only crying for a few minutes is encouraging.
We moved our son to full-time day care (he'd previously had a babysitter) when he was 2. He did the same thing as your daughter and was fine by the time I pulled out of the parking lot. You just have to walk out and not make a big deal about it.
Our son is now 4, and he had a horrible time transitioning into a new class a few months ago. He got very comfortable in the class where he was with his teachers. Even though he knew most of the kids, it took weeks for him to like going to the new class. Some kids just don't adjust to change well.
As you said, what are you teaching her if you take her out? She gets her way and doesn't have to learn to adjust to new environments. it's much harder to learn that as an adult than as a child.
Good luck - I think it's just a transitional period, and she'll be fine if you give it more time.