In reply to Denni: If you're looking at a local wall they may have kids body harnesses to hire, which will give you a chance to see if she likes it before buying a harness. Certainly though at that age she needs a full body harness.
Remember it's cold in the climbing halls in winter, so wrap her up well with a thick coat to wear between climbs.
If it is just the two of you she is going to be climbing almost constantly so don't be surprised if after half and hour she is exhausted.
Kids that age have no awareness of the dangers of a climbing wall and will innocently wander under climbers on the bouldering wall, so keep a very close eye on her whareabouts when she isn't actually tied to a rope.
For get har a pair of plimsoles (like we used to use for PE), these will be closer fitting and more grippy than trainers but are cheap enough to replace when they are outgrown.