Train from York to Cambridge

This route will involve going down the main "East coast" line at 125 mph as far as Peterborough, then changing to a slower train to Cambridge.
The whole route is relatively flat, agricultural and for England not very densely populated. A lot of fields and small villages flashing past. Though an Englishman would find it very English, a foreigner would find it bland. Just pick the most convenient train at any time.
"A couple of weeks" = mid December, when it starts getting dark about 4:30.
Peterborough though has a nice city centre with a cathedral, about 10-15 minutes walk from the station.

Ely Cathedral is also on the route and worth a look - though you might have to buy another ticket for the short hop to Cambridge - or do it another day without your baggage.

Be sure to pre-book your train for the bulk (or all) the trip as pay on the day can be expensive.

www.nationalrail.co.uk for times & fares. (Might be cheaper to split ticket at Peterborough).

If you do stop off at Ely, there is a good restaurant steps from the Cathedral, called the Old Fire Engine House. We ate there quite a time ago but I see it still has very good reviews.

This is perhaps the most "American" looking part of the U.K., looks a lot like Lower Michigan or Minnesota except the color of the crops is slightly different.

I took the East Coast line from York to London last week and I thought the majority of it was gorgeous! It's not as stunning as the Ed-York route, but if you like rolling green hills and wandering sheep, you'll enjoy it. I was amazed how green everything still was.