Train from London to York

Yes. Trains to York leave from King's Cross, so first you need to take a train or tube from London Bridge to King's Cross/St Pancras.

Hi from Wisconsin,
We will do this trip April 8. Get yourself to King's Cross, then hop the no transfer ride to York. It is only a 2hour trip each way.
wayne iNWI

Thameslink trains connect London Bridge station and St Pancras International station. Out of the train and across the street to Kings Cross station and into the train heading to York.

The different stations in London serve specific regions of the country and you don't have a choice which station you leave from. Starting from London Bridge you need to get to Kings Cross station. The simplest route is, as Nigel suggested, taking the Thameslink train from London Bridge to St Pancras (the only cross-city railway route in central London) and crossing the road to Kings Cross. If the Thameslink route isn't working for whatever reason, take the Northern Line tube from London Bridge to Kings Cross St Pancras.

For the sake of pedantry, I should point out that it is possible to take a train from London Bridge to Stevenage, then a train to York.

also in the cause of pedantry I suggest that that choice is best exercised if time is of no importance...