I think this is a great idea. I would go to something like that, and I think most of the women in my office would go, too. I think it's a good idea to gear it toward women, because if you don't, women might be intimidated. Anytime you get men (even mechanically clueless men) around cars, they start swaggering and calling women "little darlin'" and in general do everything possible to imply that women can't possibly grasp the basics of auto mechanics. If you made a point of advertising it as a women-only class, you would get a lot more women showing up for it.
Maybe advertise it as "girls' night out at the garage" or something. Offer a discount if you sign up with a few friends? Advertise on Facebook, at the nail salon, at your kids' school... anywhere moms are! When my husband showed me how to change a tire, I was dismayed at how easy it was, and was embarrassed that I hadn't learned before (and then when he showed me how to change the brakes out, I got mad because I'd been spending $700 to have a mechanic do it, and it's not any harder than changing the darn tires!). I bet a lot of women do want to learn how to do this but have never had the chance to have someone explain it to them without talking down to them.
To answer your question, yes, I'd be more likely to take my car to a shop that offered a class like this. (Mostly because I'd know the mechanic a little bit, and I'd know he didn't think me too stupid to grasp the concepts myself!) What a great idea! I hope you'll let us know how it turns out.