Chloes of York was the first tea room I have ever been to. It was also the first place I went to eat in the city centre.
It is a family run tea shop. I didn't have any of the baked goods but they looked pretty good and were reasonably priced. I had a ham sandwich with coleslaw and crisps, all of which came from Yorkshire. I also had a pot of tea. The whole meal was around 6 GBP (roughly 9.50 USD). The tea was just an afternoon blend but they had a few other options. It is fresh brewed for each person and is 2 GBP for a small pot, which is about two medium cups. The sandwich was good, as were the crisps and coleslaw. The best part was that all of the food was fresh and local.
Here's the website:
Chloes is pretty easy to get to. It is located on Colliergate between St. Saviourgate and St. Andrewgate.
York directions tip: anything with "gate" in it is a street and anything with "bar" in it is a gate (in the Roman walls).