Greetings
The above posters are correct - it really depends on where you want to go. Big cities are best done by public transport, as parking is expensive and driving a nightmare, but for smaller towns or meandering around at your own pace a car can be more convenient.
Your best bet is probably to plan to use public transport where possible - and book well ahead of time for long distance trains to get the best deals - and if you need to rent a car, do so on your way out of London or other large town.
With four people a car might make life easier, but you do need to be aware that UK cars tend to be quite small, and a large car can be problematic because of small parking spaces, narrow roads and high petrol costs (>$4-5 per gallon). So you need to figure out what car will fit you and your luggage, and car seats if any of your children are of the age when they are legally required to be in a car or booster seat. And decide whether it is worth renting.
Kate