I will be flying out of San Diego to Manchester with my 15 yr old son. We don't plan to stay in Manchester, but will travel immediately from there to York and surrounds... Plan to spend 4 days up there, then make our way down to see family in the Air Force living in Bury St. Edmonds. Considering Nottingham for a visit (Sherwood Forest?) on the way south, but not sure of the travel... assuming train? Perhaps this is the fastest way getting south from the north? We will stay approx 4 days in the Bury St Edmonds area. Then will make our way to London for 3-4 days before flying out. Any recommendations to this would be appreciated.
Re "cheap" accommodations. You don't specify what you mean, but recommend Premier Inn for excellent budget accommodations. Since I first used them 3/4 years ago they have become the first thing I look for when I need accommodation in the UK.
Was in York last year and really recommend the National Railway Museum, especially with a 15 year old.
I have made several visits to the National Railway Museum over the years , it has always been a full day , and I always left wishing I had more time ( and my wife has been as enthusiastic ! ) . More than just trains and associated artifacts , it beautifully ties together railways and the development of The British Empire in the years of the Industrial Revolution . To help whet your appetite , there is an ongoing British TV series ( BBC ) now in its eighth season " Great British Railway Journeys " presented by the former MP , Michael Portillo . It doesn't matter which episode you start with , they are incredibly addictive . Here is some information about the series https://en.wikipedia.org/wiki/Great_British_Railway_Journeys and here is a place to start watching them - https://www.youtube.com/results?search_query=michael+portillo+great+british+railway+journeys
If you plan on staying in York for 4 days, then find a bus tour out to explore the surrounding area or rent a car and explore the villages around the Moors. There is so much to do and see in this area. If you are into steam trains there is the North Yorkshire Moors Railway - leaving Pickering and heading to Whitby. The National Railway Museum is also wonderful (in York) and worth a visit. (its free to enter with an optional donation). There are also small bus companies that do bus tours out of York if you decide not to rent a car. Castle Howard is also a great stop near York. The countryside between York and Bury St. Edmunds is not very interesting IMO. It will take about 4-5 hours by car. I have not stoped by Sherwood Forest, but I can talk about Bury and the surrounding as I have been there many times. I am sure your family will have ideas on what to do here - but do go to Cambridge and try punting on the river. Also explore some of the Suffolk villages such as Lavenham. Have a great trip. I am heading to Suffolk in May.
Check out the London House hotel in London for your stay in London. I am booked there for later in the year, have not stayed there yet but it was very reasonable. It also is in the RS guide book.
Once you get to Suffolk, it's a lot better to have a car. There have been some recent posts on tge sights, on this board.
Manchester Airport to York can be done by a direct train service or you may have to change. If you can manage a day in Whitby, this is a great place to visit on the coast.
Sherwood Forest is over 20 miles north of Nottingham, closer to Worksop. From Nottingham, it's about an hour by bus and easier with a car.