I went to the SSR spa last December, and it was wonderful! I've only been to local spas a handful of times, but they were all very similar, of course, Disney's was better because of that Disney magic!
There is a private dressing room, this is usually an open area. There are sometimes people just letting it all hang out, but I usually change in the bathroom. Depending on what spa services are being done, you may just need to take off your top (for a facial) and then there is a wrap you would put around your top, leaving your pants/shorts on. If you are having more "body" treatments (massage, wraps, etc.), you would have to get fully undressed, and then there is usually a full length robe that you would wear.
For her first time you might just want to start with the basics, like a manicure, a pedicure and a facial.
Have fun - it'll be some great mother/daughter time!
