The easiest route from the east of the city to the centre is the Foss Islands railway path. This starts from Osbaldwick and Tang Hall, with a new bridge over the River Foss to link it to the Minster. It’s direct and fairly fast, though the stop-start path through the retail park is exasperating. Watch out for broken glass.
Crossing the city centre isn’t straightforward. You can head for Bootham and then south to Scarborough Bridge, which avoids the busy central bridges. From there, a cycle path skirts the station before reaching the quieter paths and roads of the western suburbs.