If you aren't familiar with Atlanta, I'd spring for the tour. I think somebody on the forum did that not that long ago and raved about the experience.
The Underground is in the Five Points Area and is "underground" - though it has an upper and lower level. It's historic, and cool and artsy and gangland all at the same time. It has funky shopping and retro style museums, and lots and lots and lots of bars - can party until 4am there with you being able to drink and walk from bar to bar. However, a lot of historic stuff has been burned to the ground and rebuilt ever since the Civil War. When I was a kid, the place was nothing but homeless people in the 80s. In the early 90s (Rodney King) the area was burned to the ground and riots were all over. It's mostly tourists during the day and partyers at night and what my dad would call "degenerates" at all time of the day.
It can be a little "iffy" if you aren't comfortable with the area, and there was a shooting there just last week. There are so many other beautiful places to see in Atlanta, I just wouldn't send a first-timer to that neighborhood. I also don't know how old your daughter is, so since the bar scene is the number one business around there, I just wouldn't recommend it. However, I did party there when I was younger.
Buckhead is lovely. Nice homes, nice shopping, nice scenery still - even in a city. My grandmother said it used to be called the Heights when she was a little girl. :shrug:
But Midtown is where it is at. Theaters, symphonies, museums, Piedmont Park, Peachtree Street aka the Strip, the Botanical Gardens, etc.
Oh and I don't know how long you will be there, but most places you can access via riding the rails - MARTA, and if I remember correctly, a lot of places will run free shuttles back and forth. Check out the places you want to hit and see if they line up with a MARTA map and offer a shuttle.
http://www.itsmarta.com/