Hi
@icowden. I've done quite a lot of riding in the Forest of Dean, all on a road bike (albeit one with bomb-proof tyres). Nearly all of my rides have been on Sundays, when the traffic has been light. Some of the A roads that cross the forest can get quite busy with traffic on weekdays.
Do look up the "Pedalbikeaway Centre" near Cannop/Coleford. By all accounts there are some great mountain bike trails there. Our club sometimes does a cafe stop there on rides out from Gloucester. These rides all involve a fair bit of climbing to get into the forest, including some killer climbs such as St White's Road, Cinderford, the Lydbrook Valley, the climb from Newnham on Severn to Littledean, and the climb from the West up to Speech House.
There is a "Forest of Dean Visitor Centre" at Soudley, which also has a cafe, and the area between there and Soudley Woods (Fancy Road) and the lakes at Cannop Ponds, and Mallards Pike is always lovely to ride.
From somewhere near Pedalbikeaway, there is an off-road track leading to the outskirts of Cinderford that makes for a great little there-and-back family friendly ride through the woods where you might be lucky enough to spot deer.
Another great family friendly off-road track runs from near the road bridge in Monmouth (a little lane where you might be lucky enough to be able to park) through the woods and along the banks of the Wye all the way to Symonds Yat. This always forms part of the "Once More Unto Agincourt" 100km audax from Tewkesbury, and it is a real highlight that can definitely be done on a road bike, though pot holed and slippy in places. I can recommend carrying on as far as Goodrich Castle for further views of the Wye Valley. EDIT: MUCH EASIER TO PARK (AND FIND THE TRACK) AT SYMONDS YAT.
Not knowing your kids' ages or ability levels, I would advise you to have a good drive around to asses the traffic conditions and the hills.
If you want to do a ride on your own, I can thoroughly recommend a loop taking in the long, sweeping descent of the A4136 into Monmouth, followed by the track along the Wye to Goodrich and a climb back up into the Forest via Lydbrook.
While you are in the Forest, apart from the cycling, make sure you take in the views of the Wye from Symonds Yat Rock and of the Severn from Littledean.
Have a great stay.