There are plenty of cycle routes around York and lots of quiet roads. You can use these to link up to NCN 1 which is a major cycling route which will take you all the way to the outskirts of London.
All the NCN routes are pretty well signposted and are either on traffic free trails or quiet roads so make for pretty good cycling. They are not, usually, the most direct routes as they tend to avoid busy main routes and visit some of the prettier parts of the country.
Maps of all the main routes are available from Sustrans. It's also a good idea to contact the local tourist offices as they will often have a range of free cycling route maps.
York is one of the UK's most bike friendly cities. It has good rail connections and also a very good Youth Hostel.
Hope this helps