Cycle trailer courtesy of London Cycling Campaign
A good place to start learning about all the options and what you will need to get up and running (or pedaling, rather) is the London Cycling Campaign's website, which has a section on riding with children and a comprehensive list of the accessories you might want to consider. The London Bicycle Tour Company hires out adult bikes and child seats on the South Bank, making it a good place to spread your cycling wings before going further afield. At only £4 an hour, it's quite a bargain as well. On Your Bike, near London Bridge, rents kids' bikes that would suit an older child able to ride by herself.
If you're reluctant to ride with your Little Londonist in motor traffic, Wandsworth-based Go Pedal! hires out child trailers specifically designated for cycle paths and city parks. So if you fancy pulling little Johnny or Susie behind you instead of attaching a seat to your bike, you can pop one of these handy gizmos on the back and go for a cruise. Just beware of steep hills, as more than one enthusiastic trailer-pulling parent has found themselves frantically struggling to get up a sharp incline only to careen wildly down the other side, trailer bopping along at top speed. The kids will probably love it but it might get your heart rate up a little more than anticipated!
Also consider checking out London Recumbents, situated inside Dulwich Park. They specialise in family cycling, hiring out a wide variety of child seats, tandems, trailers and seriously cool-looking tricycles (including ones for adults), mainly for use inside the park. Helmets for the tykes are advised for all types of cycling so be sure to hire or buy one before setting off, whether on a lovely country lane or through heavy traffic.
If you end up loving family cycling so much that you want to do it on a regular basis, you can purchase all the gear at places like Simpson Cycles or larger chains like Halfords. Better yet, visit your local bike shop. They'll be able to kit you out and give you advice on the best places to ride in your immediate area.