Hi. Newcastle is my home town, and I live in Yorkshire, so I know the area well.
Get the train to Newcastle from Manchester Airport it's around 2 and a half hours. In Newcastle you will want to see the castle, the city walls, especially the West Wall, which is in Newcastles vibrant China Town district, and St Nicholas Cathedral just by the castle. Visit the Quayside to see the famous bridges, and walk across the Millenium Bridge to the Baltic Gallery of Modern Art (it's also free to get in). Even if you don't like Modern Art it's worth visiting just to go to the rooftop viewing gallery. Lunch at the Pitcher and Piano on the northern end of the Millenium Bridge is worth it just for the view. Get the train or a bus to Durham, or even a taxi, it's not far. The cathedral is stupendous, and the location breathtaking. It's also worth visiting nearby Washington and seeing George Washington's house for some American history. Back in Newcastle hop on the Metro to Wallsend and visit the Roman museum at Sedgedunum. Wallsend is at the end of Hadrians Wall, and there is a very good visitor centre there.
Hop back on the Metro and go to Tynemouth. Postcard pretty, with stunning sea and river views, and long sandy beaches with some very hardy surfers always there. Some great little restaurants in Tynemouth Village, notably Allards, or take a walk to the nearby Fish Quay for superb fish and chips at Christians. There is also a very good cafe at the southern end of Tynemouth beach, right on the beach under the Grand Hotel.
Regarding food, there are some great value restaurants. Santana's in Jesmond do great pizza and pasta, for £5.95. In the city you will find most of the usual chains. Head for the Bigg Market for a curry, Chinatown for Chinese Food, or the quayside for smaller, more quirky places, but Newcastle is a real party city, so it's very lively on a Saturday night, especially as Newcastle United are playing at home on 4th May (v Liverpool) and 50,000 thirsty supporters leave the ground to celebrate or drown their sorrows. The Geordies (Newcastles inhabitants) are incredibly friendly, so expect to be the centre of attraction.
Go to York on your way to Nottingham. Lots to see, but first job is walk around the city walls. National Railway museum is superb, the Minster is astonishing, and the general ambience is relaxed and prosperous. Try the "Go Down" restaurant near Clifford's Tower, and have a beer in the Kings Arms on the riverside, the most flooded pub in England. If you want something different go to Selby, (15 mins by train) and visit Selby Abbey to see the Washington window with the Washington family coat of arms, considered to be the inspiration for the Stars and Stripes, and an fine example of an abbey church. Some nice little restaurants in this pretty, small English town, at out-of-town prices.
There is not a lot of Nottingham Castle to see, but the city has some interesting history, and the smallest pub in England. Jack Straws House is worth a visit as a perfectly preserved 1930's workmanship house.
Lincoln is very beautiful, and the cathedral tower is the highest in England. It's a sort of mini-York, with fewer tourists, and is worth a visit.
In Manchester do the Science Museum. Manchester saw Rutherford split the atom, Frank Whittle invent the jet engine, Allan Turing invent the worlds first programmable computer, the industrialisation of cotton spinning, and the creation of Rolls Royce so it's worth a visit. Also go to the Manchester Art Gallery especially to see their collection of Pre Raphaelite works.
Liverpool is a half hour by train from Manchester. See the Cavern Club and hear tribute bands play where the Beatles once did, do the double decker bus trip, and get off and have a beer in The Philharmonic pub (John Lennon said "the price of fame is not being able to have a pint in The Phil"), and walk around the docks.
Have a great trip.
