There's a fine little loop trail, formerly known as Ski Loop #1 in the Dundas Valley, now labelled as Deer Run Trail. The path encompasses many of the joys of the valley, with steep ravines, lovely forest cover, and well defined trails. Of course, the aptly named Deer Run provided live specimens of the deer-y kind (below).
If longer hikes aren't on your agenda, or you have little ones who want a taste of the forest without an epic hours-long adventure, this is a great introduction. For little children, in fact, this would be epic, with lots of up and down hills, curving pathways lined with tree stumps and branches (pictured below) to help them keep on the trail, and if a quiet interlude can be maintained, there are bird songs and possible animal sightings to seal the deal.
This trail is very close to the Dundas Valley Trail Centre. To get there from the trail centre, walk across the Hamilton-Brantford Rail Trail at the Main Loop Trail, and take the first right, directly at the top of the hill. Do the full loop and reward little ones (and adults too) upon completion with affordable treats (chips, granola bars, etc.) food (hotdogs) or beverages (pop, juice, energy drinks, fair trade coffee) at the Trail Centre during their regular hours.
I suspect too that this trail is not as busy with mountain bikes, which always are a worry to parents with little ones on some of the longer through trails.