Firstly you will love the Cotswolds, it's such a pretty area, cobbled streets small houses with thatched roofs, solid oak beams that run through the ceilings.
The Cotswolds are made up of many towns, places like Stow on the wold, you will see many beautiful things in and around that area.
If you think of the shape of England it's like an L shape, the distance from east to west is approximately 430 miles
From direct south to the north, in Scotland is approximately 600 miles.
The two furthest points are from Lands end in Cornwall to John of Groats in Scotland which i believe is about 720 miles, there is a sign post stating the distance, but i can not remember the actual distance.
Although many states in America are larger than the whole of England, do not be fooled thinking you could take a relaxed day trip from the Cotswolds to Bath, the road sytems in England are very slow, they have too many circles/roundabouts and many stop signals. By the way YOU CAN NOT TURN ON RED on any light. If you do, expect your credit card to be charged about $120 each time the cameras flash.
England has many speed cameras. and they will catch you if you exceed the speed limit. Here is a general rule of speed limits for a reference, anywhere you see a house or residental area the limit is 30 miles per hour, single lane traffic is generally 40 miles per hour, two laned traffic going in the same direction is 60 miles an hour, and the motorways are 70 miles an hour.
You are going to love the price of gas/petrol at $9 per gallon LOL. Okay that is the bad news out of the way, but England is a beautiful place to visit, Bath is a great place, and would probably take you about three hours from the Cotswolds.
Cambridge is really worth a visit, and not that far from the Cotswolds, the river Cam runs through the town, and it's great to hire a punt and take a trip down the river, you will see many of the buildings that make up Cambridge university, they are very impressive. If you headed North east from the Cotswolds and drive round the Norfolk Broads, In Norfolk these are inland waterways that are very pretty and you will see the long narrow boats and see how the lock gates work within the canel system.
If you plan on going to London, I would suggest that you get close to the City then take the underground,tube trians, London has a lot to offer, but can also be a nighmare trying to drive around, especially as a tourist, the public transport in London is excellent, you can purchase an all day ticket and just get on and off the tube anywhere to explore.
You must see some of the coast of England, Places like Rye harbour are worth a few hours, its a very old traditinal fishing village, but over the years, the streets have become very popular with the tourists, the streets are very narrow, and they have many quaint little shops and places to eat and take a break.
You mentioned about the Lake District which is definately worth seeing, but to travel each day from the Cotswolds seems like a lot of driving.
The county of Kent is very lovely that is east of London on the coast, it's actually known as the garden of England, there are many apple orchards in Kent, and it's also very famous for their hops which they grow to produce some very fine beers and ales.
Here is a list of some towns and places that I would say are worth a visit.
Brighton it's in east sussex, very cosmopoliton, it's on the coast and has some excellent stores, a place in Brighton called the lanes is a good place to purchase unusual items.
There are countless shops, stores, wine bars, pubs and restaurants in Brighton.
The lanes are just off the seafront, in front of the pier.
Eastbourne in east sussex, is also a very nice town, it's not as large as Brighton, but has a more of a family and local feel to it, the seafront is much better than Brighton and they have sandy beaches all around the pier area. Eastbourne is known as the sun trap of the south, and there is a lot to do there. A nice walk from the pier going towards the cliffs is nice, the cliffs are very famous, they are approximately 600 feet high, and there is a fully operational light house beneath the cliffs.
I live in the USA now, I left England about four years ago, although i am happy here, I do miss the English countryside, and the coast, I have not seen a single castle since my arrival to this county, but you will see many in England, Heaver castle is a popular place, it was the home of Henry the V111
Okay that is some little bit of information for you about England, If you get the chance or opportunity, head west, see places like Devon, and Cornwall, they really are beautiful.
I wish you luck on your trip, I know all of my American friends who have been to England have had a great time, and some think I am crazy to give that up to be here, but a vacation is so much different than reality.
Paul M