Hope I'm not too late with train ticket info... because this trip looks like two weeks of fun and I wish I was doing it too... HOWEVER - you have set yourself a LOT. If you chose to drive this, it would take MUCH longer so I recommend train as the way to go. You know cars, but this is England.
There are train tickets you can buy OUTSIDE the UK for travel on Britain's railways nationwide, and one you can get in the capital for travel within London on public transport.
BRITAIN - the BritRail pass is US$707 if you use it to travel on 8 days within a calendar month of first activating it (it's the Flexipass option).
https://www.britrail.net/passes/britrail-pass
GET THIS PASS ASAP AND RUSH SHIP IT!
On day 1 you could get the train to Salisbury and then the bus from Salisbury train station to Stonehenge and back again, which lets you enjoy the rolling hills of Wiltshire.
Day 2 and 3, use the rail service to go to Stratford Upon Avon for a dose of The Bard, and to Moreton-in-Marsh for the Cotswolds. There's a day trip you could go on with a local driver and guide. This will give you the authentic local experience with no hassle. At the end of Day 3, that train to London Paddington (trains in the evening at 6.11, 7.40, 8.44, 9.41, with a final one at 11.22pm. Journey time just under 2 hours and the train terminates at Paddington so you can snooze a while). HOWEVER, it does take over 3 hours to get from Salisbury to Stratford on Avon (changing trains at Basingstoke and Leamington Spa).
https://visitbytrain.info/cotswolds/index.htm
Then the London bit is below. It's shorter because I just talk about the best ticket to get.
Side trips by rail on the other days - trains to Liverpool leave from London Euston and take around 2.5 hours each way, and then a side trip to York will add an additional 2+ hours journey from Liverpool (as you'd go through Manchester and Leeds, the width of the north of England!) so for that day you might want to consider a triangle trip (Euston to Liverpool in the early AM, mid-day train to York after a few hours Beatles sightseeing, then a late evening train back to Kings Cross which takes 2 hours in itself).
Trains to the White Cliffs Of Dover take 2 hours as a commuter train from London Victoria direct, or 85 minutes if you get the super rapid train from St.Pancras to Ashford and then change for the half-hour ride between Ashford and Dover Priory.
The trains to Cambridge leave from just across the street from St.Pancras at the suburban platforms of King's Cross ...which means you'll see Platform nine and three-quarters (fastest trains to Cambridge take under an hour)! You are planning to go to Cambridge on a Sunday... there may be engineering work on the line, and not as much open, so maybe switch the Cambridge trip to the Monday May 14th. That way you do Dover on a Saturday (and people might be going to the seaside on that day so that's a great experience), chill in London for the Sunday, then Cambridge on Monday.
For all your rail timetable (schedule) needs, National Rail website is the best! That's where I planned all this out.
http://www.nationalrail.co.uk/
LONDON - buy an Oyster card for £5 when you're there. It's an electronic transport card with an RFID chip inside, you load it with payment and the payment is deducted off the total as you scan in and out of stations. It's very 21st Century. For the six days you're there, I recommend putting £50 credit on it. If you just travel in the center, you'll not be charged more than £6.80... if you go to somewhere like Wimbledon for the day, it's slightly further out and that's £8.00 before the card stops deducting further amounts for the day. Use it on buses, the underground, local trains in the local area. You'll love it, traveling around like a local and seeing all the sights.
https://www.londontoolkit.com/briefing/oystercard.htm
