Cycling with kids -
Get in touch with you local branch of the CTC (Cyclists Touring Club, but now about an awful lot more than just touring). I can;t find a list of local branches or activities on the main CTC website, but the Head Office contact details are here. Give them a ring/e and they'll tell you who your nearest branch are. Many branches do family rides, kids cycle training, women only rides as well as more gnarly stuff.
Article on Cycling with Children here.
CTC are lovely, they are non-competitve.
If you want to go it alone as it were, we ride with our 2 year old on Sustrans routes and in Forestry Enterprise forests. Both provide easy, flatish, family friendly routes. Some Sustrans routes are old railways, others are waymarked bridleways and quiet roads.
In Scotland, the 7 Stanes forest parks have bike routes for all ages and abilities. In Wales, Google on Coed-y-Brenin (Dolgellau), Marin Trail (Betws y Coed), Afan Forest (Cardiff/Swansea way), Cwm Carn (SE Wales, closest to England) and Brechfa (Mid Wales) do something similar. Look here.
If you are in the M62ish corridor, then the TransPennine Trail is a good place to start. We use the old train line around Holmfirth - Penistone - Sheffield a lot.
If you can get to Nottinghamshire, we have a lot of days out to Sherwood Pines Forest Park and the National Trust estate at Clumber Park. they both have waymarked family rides, cafes and playgrounds. Might be worth checking your local Forestry Commission and National Trust estates.
Forestry Enterprise cycling Centres in England
Closer to London there is Swinley Forest (Windsor) and <a class="break-all" href="http://www.gorrick.com/swinley/www.cityoflondon.gov.uk/Corporation/LGNL_Services/Environment_and_planning/Parks_and_open_spaces/Burnham_Beeches/visitor_information.htm" rel="nofollow" target="_blank">Burnham Beeches. Maybe also Epping Forest? It's been a long time since I lived in London (grew up there), but also Richmond Park - we did a lot of family bike riding there. Wandsworth, Wimbledon and Tooting Commons too, and Putney Heath.
As for how to transport kids - our experience
DS (very nearly 3) went in a trailer at about 6 months - this was a bit young really, and he could manage about 20-30 min spins at a time.
When he was about a year old, we got a baby bike seat - A Hamax Kiss - and he loves that, he can see so much more. He and DH go out for hour long rides or so, maybe a bit longer, but DS loves to run around so he gets a bit bored. He got a balance bike for his 2nd birthday. We splashed out and got an Isla Bike Rothan. It is superbly well made and worth every penny compared to my friends' much cheaper one, which they've had for the same amount of time and is knackered already. DS is a demon on his balance bike and likes to ride to nursery most days. He's really got the hang of balancing and he'll move up to a proper bike before long - maybe 4th birthday.
For competitive kids, British Cycling (the gov skills and development body for cycling) run an initiative called Go Ride, which organises race training, skills development, kids activities and so on.
Triathlon clubs also often have a family section which organise bike rides. Look at British Triathlon but be prepared to be talent spotted!
Sorry this is all a bit mixed up, but I was trying to get all my thoughts down before I forgot them.