I live in York.
£600 to £850K doesn't get you a huge deal in York compared to other areas of the UK. We recently viewed a small-medium 3-bed detached that needed everything doing which was up for £650K and went, we're told, significantly over asking price. It's a sellers market in York.
In terms of areas, people will be along shortly to tell you about South Bank as though its the answer to everything 😂
I, personally, find SB full of middle-class, lentil-weaving wankery. People who read The Guardian and want you to know that they read The Guardian.
You'll find similar properties cheaper in Holgate which is about the same distance from the station. People in Holgate probably do read The Guardian but don't really care if you know it or not.
The area around Bootham (over the other side of the river) is very nice. It's similar to SB - rows of Victorian terraces and also quite expensive. But its closer to town and the station.
Fulford has a more varied housing stock but its a little further out and half hour might be a bit optimistic for a walk to the station.
Tadcaster Road area and Dringhouses has some beautiful houses and quite varied stock. I'm thinking of the area around Hunters Way, Middlethorpe Grove and The Horsehoe. But the houses can be quite large and, therefore, quite expensive so you might find yourself priced out.
I'd say avoid Lawrence Street, Heslington Road area because it's all York University students. The Groves and the areas off Bootham 'behind' the hospital is all York St John's students and trainee medics. People on MN are very negative about Acomb and Tang Hall because they are large post-war council estates with associated deprivations and problems. With a budget of £600K, I guess you wouldn't be looking in those areas anyway. But, don't write off the whole of Acomb. The area around Acomb Green and beyond (towards the A64) is very beautiful and only about half hour to the station.