I went to university in York, and it's a wonderful place, beautiful and absolutely steeped in history. The main part of the city is still surrounded by the old Roman city walls, and you can walk most of the way round the city on the walls. The gates in the walls are known as Bars, and many of the streets are called Gates. One of the shortest streets in York has one of the longest and most curious names - Whip-ma-whop-ma-gate. In the spring, the slopes at the bottom of the wall are thick with daffodils.
Bettys Tea Rooms is a mini-chain which has six branches across Yorkshire. The main tea room that most people visit is on the corner of St Helens Square, in the main shopping area. It's pleasant enough, but it's a very busy corner and the place has huge windows which make it feel a bit like a goldfish bowl - if you're lucky enough to get a seat on the main floor. It also tends to get very long queues, and at times you have to wait a good while for a table.
The big tip here is that Stonegate, which is a lovely cobbled street leading up from St Helens Square to the Minster (another place that mustn't be missed if you're in the area - and if you can, take a pair of binoculars with you when you go in and don't forget to look up at the ceilings and upper windows), has another branch of the same tea room, called Little Bettys Cafe. This place is a much more unspoiled old building, with the main tea room upstairs. It serves exactly the same things as the main Bettys, but usually has much shorter queues and a better atmosphere.
The Castle Museum is a museum of everyday life, and is one of the best museums I've ever visited. It takes you from Roman times right up to today, and as well as the really ancient stuff, it has things like televisions and washing machines, so that you can see how household appliances developed over the 20th century. As a student at York, I was able to go in free of charge, and I went there often to browse the different collections.
The Jorvik Viking Centre recreates the sights, sounds and smells of Viking Britain and is probably worth a visit if you've got the time, but is quite expensive and tends to have quite long queues. I'd say it was more aimed at families with children than adults on their own, and the main part of the tour is over quite quickly - you go through on a sort of train and don't get to stop and look at things at your leisure.
If you're at all interested in trains, the National Railway Museum is well worth a visit. As well as the static trains that you can walk round and (many of them) climb on, there's a fully operating steam train on which you can take a (very short) ride. From the centre of town, the most pleasant way to walk to the railway museum is along the towpath by the river - I used to walk along that bit of the river every day in my lunch break when I worked on that side of town one summer, and would sit in the gardens by the City Rowing Club to eat my sandwiches.
York is just a couple of hours from London by train. A great money-saving tip is that if you book on Hull Trains from London to Selby, it's considerably cheaper, and you can then take a bus from Selby to York, which takes just under an hour. Definitely well worth a visit if you have a couple of days available.