The best places to hire a car - with the widest choice of cars, most deals, most companies etc., will be at airports, particularly Heathrow. I'm not clear on what you mean by 'catch a train out of London and hire a car', but if you stop off at Heathrow, which is not far from central London, you can pick up a car there. There are a variety of ways to get there, although a taxi may actually be the cheapest and easiest if you have your entire family and a lot of bags.
You can also pick up cars in the centre of London (edit: don't forget the congestion charge) - most of the major rental companies have several locations - but check opening times etc. Wherever you rent from, I'd advise making an advance reservation to avoid disappointment.
If you meant to take a train out of London, then pick up a car, your options will be much more limited, unless you travel to another major city. Car hire locations are typically near airports and major train stations. The train system in Britain is generally OK, although you may find it a bit expensive. Travelling to rural attractions without a car will be almost impossible however, in many cases, unless you are prepared to spend a fortune on taxis.
I don't see any major practical problems with your plan. If you are used to LHS driving, you should be fine. You can rent an automatic if you need it (make sure you make that clear with your reservation as manuals are normally the 'default' in the UK), although manuals will give you more choice.
If you are driving around small towns, remote areas, etc., it's worth getting a guide to Bed and Breakfasts (aka guesthouses, typically in a private house) in the UK. In March I would expect many won't need advance booking, but it's definitely still worth calling ahead on the day you might plan to stay.
Edit: one more thing - although signage in the UK is generally excellent, roads are narrow and twisty in many places, so you'd be advised to get a GPS unless you are confident that you are travelling with a strong navigator and a good map. Saves on arguments in my experience :) Most major rental companies will rent you one, of course, although it might be cheaper to make your own arrangements.
Enjoy your trip!