The tourist office has a useful free map showing York's cycle routes, or visit iTravel-York (www.itravelyork.info/cycling). Castle Howard (15 miles northeast of York via Haxby and Strensall) is an interesting destination, and there's also a section of the Trans-Pennine Trail cycle path (www.transpenninetrail.org.uk) from Bishopthorpe in York to Selby (15 miles) along the old railway line.
You can rent bikes from Cycle Heaven at the train station for £20 per 24 hours.
Local bus services are operated by First York (www.firstgroup.com/york). Single fares range from £1 to £3, and a day pass valid for all local buses is £4.50 (available on the bus or at Park & Ride car parks).
Station Taxis has a kiosk outside the train station.