I take it you will be landing in London - possibly Heathrow or Gatwick, which is the west side of London. So you may as well explore the area around Hastings, Cantebury and along the south coast to Brighton (buzz city, with pier, beach, and fabulous shopping), from there continue to follow the coastal roads across to Chichester (beautiful city with cathedral and quaint shops - even MacDonalds is in a gorgeous old building)...then head up north west to Winchester (another fantastic city with beautiful buildings, see the castle ruins, meander around shops, bistro's, small cobbled streets). From there go west to Salisbury, and up to Amesbury and pay a visit to the most famous stone structure in the UK, Stonehenge...then further west to Cheddar Gorge, Bath, Glastonbury, Wells (smallest Cathedral city in England) If you have transport, time and money...I would drop back down to the south coast from here, and visit Lulworth Cove and perhaps along to Salcombe Bay for some of the most stunningly beautiful coastlines the southcoast has to offer. Head from there to Taunton (cider, and quaint rural scenes, rustic charm and friendly people).
Head up through Bristol (university city, visit Clifton) then up towards Gloucester, then head east along A40 towards Oxford, passing through the beautiful suburb of Witney and across some fantastic rural areas into Oxford (university, river, great shopping).
From Oxford head north up the M40 towards Stratford-Upon-Avon and visit Shakespeare country, see some beautiful buildings and incredible scenery, and up into Warwick.
From Warwick I would catch the train or National Express bus up north to Stafford, and see where the invisible line between the north and the south divides...listen to the local accents, where they pronounce words without adding extra 'r' to the words and thus confusing the tourists!
Up in this area you will find Alton Towers...the best theme park in the UK, take time out and spend a day going on all the brilliant rides here. Nearby there is a fantastic Youth Hostel in an old building with heaps of character and local charm. And then head up into Cheshire, and see the market town of Macclesfield, head into the Pennines and visit Buxton.
From Macclesfield you can catch the train into Manchester Piccadily station, and step out into the heart of Manchester, the industrial city of the north. Visit Piccadily gardens and join the tram for a decent tour of parts of the city. Visit the famous Arndale Centre, and find some truly individual shops (nothing like you will find in the south) and experience the cinema for bargain prices, and see where Coronation street is filmed. Visit the famous Manchester ship canals, and see the old mills and charm that Manchester has to offer. Stay in the brilliant vibrant Youth Hostel. And visit the Lowry Museum in Salford Docks.
From Manchester you should head north to Blackpool, Lancaster and up through into the Lake District, taking in Ambleside, Kendal, Penrith and up to Carlisle. Then further north still to Gretna Green, and up through Dumfries and Galloway to see some of the most incredible landscapes you will have ever seen, for miles and miles and miles. Carry on right the way up into Glasgow and see the beautiful industrial landscape and visit local pubs and see if you can understand the Scottish language and have a dram or two of whiskey.
Head further north to Stirling, then across to Fort William, then up to Inverness - I think once you've got there, you will want to travel up and up some more....from Inverness to Aberdeen and down the coastal route to Dundee and across into Fife, and across to the most famous Golf Course in the world...St Andrews...check out the university (where Prince William studied) and thendown past Kircaldy and onto the Firth Forth road or rail bridge across towards Edinburgh - the capital of Scotland. Spend a good day or two exploring this beautiful incredible city, steeped with history and tradition. Then head of east to the coast line of Berwick Upon Tweed (you wont be disappointed at the stunning coastline here).
Down to Newcastle and make sure you spend a night out here, Newcastle is usually bristling with happy party-goers, there are plenty of bars and clubs, and prices are a fraction of the costs of London....and the atmosphere is so much friendlier than any southern town at night.
From here to the beautiful understated city of Durham...check out the university site here, its incredible and picturesque. From here to Harrogate (spa town of the north)...again steeped in history and picturesque. From here across to York, visit the museums and walk around the wallls of the city, and see the Viking exhibitions and learn the history of the area.
From York across the Humber Bridge (opened by the queen in 1981) and down to Lincoln, from Lincoln across the Lincolnshire Wolds to Skegness and see a traditional british seaside holiday place which is still alive and happening today. From here travel down the coast, and take in Boston, Horncastle and see Tattershall Castle, across to Peterborough (you wont be disappointed, its beautiful too). Then down into Cambridge where you must take time out to explore. Travel across country into Suffolk, seeing Sudbury and Flatford Mill and Lavendar and see Constable Country at its best.
From Suffolk head into London and explore the capital, making sure you go to Camden Town, Soho, and Carnaby Street, Oxford Street and take in a theatre show or musical.