Duncombe Park, Yorks.

The house, flanked by two large square service wings, is set in extensive parkland. The house was completed in 1713 and remodelled in 1843. The central block was gutted by fire in 1879 and not rebuilt till 1895, with a slightly revised layout.
I visited Duncombe Park in 2005. Inside, there is the Stone Hall, a lavishly decorated forty-foot cube, and other fine rooms with period furnishings. The house is no longer open to the public, but you can still visit the park and gardens, which are open several days a week during the warmer months.
(If you are curious to see what the house is like inside, check Weddings/Ceremony Locations on the Park website, or enquire about the Special Events.)
The gardens are quite extensive and include two temples, a large formal terrace, and a ruined conservatory.

Cliffords Tower, York

The tower comprises an unusual quatrefoil shell wall, built on top of a high mound. It’s almost all that remains of York Castle. Visitors can climb to the top of the wall, which affords great views over the central parts of York. The middle of the tower, which may once have been roofed, is now an open courtyard. If I remember correctly, there are a few relics and exhibits to look at inside the tower.
If you’re disabled, check photos of the mound and tower before planning your visit 🙁
There is a pay car park nearby, but you probably don’t want to be driving around the centre of York during the day, and you may find access on foot or by bus less troublesome.

Castle Howard, Yorks.

One of the great stately homes of England, set in a broad estate dotted with monuments and other buildings.
Most of it was built between 1699 and 1712 for the 3rd Earl of Carlisle, to a design by Sir John Vanbrugh. The south or garden front has two wings on either side of a central hall and dome. The house was badly damaged by a fire in 1940, which destroyed the dome and twenty rooms. Some rooms have been restored over the decades, but the east wing remains a shell.
Today the house contains many treasures, including several important collections of pictures, porcelain, sculpture, an enormous wine cooler, and the William Morris screens.
Castle Howard’s other claim to fame is that it was used as the location for the fictional “Brideshead Revisited” TV series. Some fire-damaged rooms were part-restored for use as sets, and others were restored with TV money. An exhibition about the filming can be seen in the house.
Outside there are gardens, and a number of listed monuments, including the Temple of the Four Winds, the Mausoleum, and numerous statues. There is also a huge lake.
There is enough to see and do to occupy the visitor for much of the day. If you want to see the monuments it helps if you like walking.

Beningborough Hall, Yorkshire

The great baroque house was built in the early 18th century. Notable are long vistas down the corridors, and the high quality of woodcarving on friezes and overmantels. The contents were dispersed in a 1958 sale, but some have been bought back since. The house also hosts a loan of 120 portraits from the National Portrait Gallery, and some porcelain from the Ashmolean Museum. Outside, a garden lies along the south side of the house.

Yorkshire Air Museum

The Yorkshire Air Museum, situated on a former WWII bomber base at Elvington, lays claim to be the largest independent air museum in Britain. It features 15 exhibitions, 50 aircraft, many of them in working order, and a large range of military vehicles. Also commemorated is the wartime presence of two French heavy bomber squadrons – the only two to operate from Britain. Aircraft in the collection range from WWI replicas to a Nimrod. Airplane buffs should allow at least half a day for a visit.

York Railway Museum

Also known as the National Railway Museum, York.
For a train buff, heaven would probably consist of a day here, plus a ride on a steam-hauled mainline excursion train. There are about 100 locos, 200 other items of rolling stock, and hundreds of thousands of other objects, everything from rubber handstamps to turbine powered railcars.
The museum has free admission, and is a couple of minutes’ walk from the York mainline rail station. It’s possible to walk to the museum from anywhere in York town centre in about 10 mins. There is a car park in Leeman Road adjacent to the Museum. If you think the daily parking charges are a bit much, you could take a bus from your accomodation to the centre of York, or use the ring-road park and ride from Rawcliffe Bar or Askham Bar, from where services 2 and 3 should drop you outside the museum. Central York during the day is not a place you want to be if you are a car driver. (I was staying in the suburbs of York and came in by bus).
BTW, there is another branch of the NRM at Shildon, Co. Durham. The collection there includes the Advanced Passenger Train.

National Museum of Scotland, Edinburgh.

Some years ago the museum acquired an extension at one end, marked by a new tower entrance. The new bit had the History of Technology stuff and there was a lot to look at in there. The trendy architectural layout was a bit confusing. Now I hear that the old bit, with the galleried hall, has been given a make-over. A major destination – it’s probably impossible to see everything in one day. Suggested visit duration: as much time as you feel like devoting to it.