Hi there. These are my recommendations for Dallas shopping (I've lived here over 10 years).
1.
Galleria = not real close to downtown, probably 20 minutes away, has many upscale department stores and boutiques (Saks /
Nordstrom / Louis Vuitton /
Gucci), very huge mall with ice skating rink!! If you're a tourist, this is definitely the mall to go to!
2.
Northpark Center = I personally like Northpark better. It has Neiman Marcus, a more upscale
Nordstrom. Other stores in Northpark = Burberry. No LV or Gucci. Is a much smaller mall than the Galleria but I'm a Neiman's addict so that's probably why I like it better. Is closer to downtown... not even 10 minutes away.
3.
Highland Park Village = this is the mecca of high end stores! Hermes, Escada, Jimmy Choo, Chanel. This is also fairly close to downtown... probably about 10 minutes away.
4. There is a Neiman Marcus downtown, which I think by far has the best selection of designer items. Of course, it's downtown so close to where you're staying but I would still cab it because parts of downtown are scary.
This is all I can think of right now, but I will add more later.