In reply to flopsicle:
The simplest and quickest way to get going would be to go to Horseshoe Quarry.
There are some routes there that, on the upper tier (Dr Who names, sector name escapes me now) that you can lean over to reach the lower off from above. So you can set up a top rope very easily. They're low but enough to keep a little one entertained all day, to make it worth the walk up.
Getting the right gear is going to cost a bit, a clip stick or trango spider can help enormously.
The problem for little kids is that the easy sport stuff that's available often has unfriendly (reachy) starts and that is always going to be a bit of an issue.
Easy trad climbs are generally much friendlier at the start ( and after as well) but you're going to need more training and more gear, so more expensive.
Nottingham Wall are doing the subsidised trad courses again this year (£100 for 2 days IIRC? £60 for 1 day)so that would be a good start.
Paul (Atkinson?) from Peak Mountaineering does good intro courses but I think it might be a bit late now, as he does them in April and May, he advertises on here.