Hi, I am lucky enough to live near York, and of course there's plenty to keep you going for a month, let alone a week. So here's some info from a local.
You don't mention when you were visiting. Although places like Fountains Abbey open all year, others don't. Castle Howard, Harewood etc are 'put to bed' usually from the end of October to Easter. So do check on the websites if you are visiting out of season.
My hit list would be (in no particular order)-
1) York (2 days easily to see everything)
2) Castle Howard
3) Yorkshire Dales (walking, scenery, waterfalls, abbeys)
4) Yorkshire Moors and North Yorks Moors Railway.
5) North Yorks coast - Whitby, Robin Hoods Bay, Staithes etc
6) Fountains Abbey
7) Leeds if you are into shopping. Some of the best shops outside London in a compact centre.
8) Harewood House
9) Holy Island
And of course you can easily visit Edinburgh by train! For goodness sake, my husband commutes from York to central London for work at least once a week! The GNER service is quick and efficient. The food is good so catch an early train, breakfast on the train and be in Edinburgh mid morning. The scenery on the way up is fantastic so bag a seat on the right hand side of the train to catch the views - the sea practically comes up to the line in places!
Hope this helps.
M

And don't miss Betty's.

http://tinyurl.com/y7gd69

Morgana and Carrybean. Have it absolutely right. York is central to Yorkshire which is apretty attractive part of the UK. Loads to do, but don't stay in York for the whole week get out and about.

I would also check the yorkshire web site for festivals and how they fit in with your visit. The cities around York have all the usual jazz festivals plus may and mela.

You know I deliberately omitted Betty's. I had some of the worst--and driest--cake in there for a very high price. I know most people like it, but, at least in so far as I am concerned, I thought it was overpriced, not very good, and totally a waste of time.

Feel free to ignore my advice. I know I will be out voted, but I would never go back to eat at Betty's again.

I do recommend against doing day trips to Edinburgh or London using York as a base. While the train ride to Edinburgh is scenic if you go along the coast, you would not have much touring time before you had to return because you would spend too much time on the train.

My personal take on the entire continent of Europe is that you could simply throw a dart at the map (which is what I do with my home exchanges in a way). Wherever it lands, there would be plenty to do within a 1-2 hour car ride one way to occupy you for two weeks with day trips--and probably for much longer. You can't go wrong with Edinburgh or York (for the attractions of the city itself, if forced to choose, I would vote for Edinburgh). There is no point in trying to cover too much ground and seeing everything on that list you have developed because you would simply get it exhausted and spend too much "schlep" time and not enough time looking. After many trips to both the UK and continent, I haven't come close to "knocking off" everything on my list--and probably never will. Max out the attractions and sites near where you are sleeping and save the others for another time.

By the way, the Borders region of Scotland-England also has a lot of interesting sites (not all of which, of course, I have seen).

Newby Hall is also worth a half a day. Also, whatever your faith, or lack thereof, make sure you attend evensong at the York Minster every chance you get. Sublime music. Plus, you'll avoid having to pay admission to see the interior of the church.

I agree that, with day trips, a week in York will be fine. However, I am not the greatest fan of the city of York itself. I like it, but it feels a bit "Disneyfied". The area of the shambles especially. I personally would not want to base out of there for more than two nights. I'd prefer to move on to somewhere in the countryside. But that's just me. Like I said, with a car to get out and around, you should have a terrific time.

I do agree with recommending evensong at York Minster. I cannot tell you in how many cathedrals I have been to evensong because I go every chance I get in the UK.

By the way, I am Jewish, but the music is heavenly no matter what faith you are.

Faux, thanks for the Betty's heads up although everything I had a few years ago was just wonderful. I hope you just hit a bad day.

My experience of many trips to Bettys is that the food is OK but overpriced and it's the atmosphere that you pay for.

We do still go from time to time, particularly with my grandparents who really like the atmosphere.

Also in York is the area called the "Shambles." You should probably have a look - see while you're there.

http://en.wikipedia.org/wiki/The_Shambles_(York)

Thank you all for giving me such wonderful and appreciated advice. I wish I was leaving today!! We will be on our trip late June early July. Thanks again!!

Whether I hit Betty's on a bad day or not, I would choose some other place to have a piece of cake the next time--so much hype and so unsatisfactory!

We loved York -- even though I grew up in England, we never visited until our daughter did her junior year at the Uni.

Best memories were evensong at York Minster, the Dales, Castle Howard, lots of pubs, nice places to eat, and (for DH) the railway museum. We did an excellent walking tour of the city walls with a local resident as guide (found through our hotel, IIRC) -- he was so engrossed in his topic that the tour lasted twice as long as advertised! After that, we were ready for toasted tea cakes at Betty's, for sure.

Train trip to Edinburgh would be feasible and the scenery is great, as others have suggested. Once you get to Waverly station, you could very easily get one of the "round the city" bus tours that leave right from the station. Good way to get an overview, and if it's a nice day, you can sit on the open top. But truly, there's so much to see in both places that you should just go back and do Scotland as its own trip. In late June, you'll have those long daylight hours -- should be a lovely time to visit.