We bought a Victorian house in Zone 3 when we were expecting our first, which had a decent sized garden, tube line (Northern) and a local park, good local shops etc. However, we moved out after 18 months, as I got so sick of not being able to park near the house with a pushchair and shopping etc. Also, we were able to buy a lovely historical property with beams and stories to tell, which I fell in love with!
Also, when we were in Zone 3, the commute into the city was an hour door to door, which was hardly any shorter than going in from Herts, and at least you could get a seat on the train, which made the journey seem shorter than strap hanging on the Misery Line!
It always amuses me how many people come onto these threads to say how easy it is to get round London and nowhere else compares because there are parks and museums, as though they don't exist anywhere else!
Trust me, there are amenities and facilities outside London! Also, it really isn't that easy to get around London quickly unless you live in Zone 1 or, at a push, Zone 2, unless you are travelling somewhere close by. North London Zone 3 used to take an awfully long time to get to the Natural History or Science Museums.
In fact, we have always taken the DC to museums and the theatre from an early age and I often think that we have done more than other parents we know who lived in London, but seemed to do very little.
The advantages and disadvantages of both have already been summarised by previous posters, but another massive plus to moving out, in my opinion, is that the DC have been able to have dogs, cats and horses, which massively enhanced their lives. There aren't too many gymkhanas in central London!
Having said that, I love London and, now the DC have left and gone to uni, we are moving back to London, Zone 1!
Why don't you try renting outside London and see how you get on?