On foot
The roads within the old city i.e. within the city walls are pedestrian precincts, closed to all cars except disabled drivers and emergency vehicles between 8:00 and 16:00, and most of the sights are only a short walk between one another. Take care walking around the city centre when the roads open to car traffic at 16:00, as the roads fill up quickly with delivery vehicles servicing local shops and businesses. The city centre is small enough to walk from one side to the other in 20 minutes.
By bike
York is one of the most cycle-friendly cities in the UK - there's an extensive network of cycle routes in and around the city, and most of the traffic controls have been set up to give cyclists priority. There are no significant hills in or around the city centre, which is a big help. The river path along the Ouse contains some wonderful bike routes out of the city. Also beware that police and CCTV operators take a very dim view of cycling without lights after dusk, or cycling in the city centre pedestrianised area before 16:00, and will happily hand out an on-the-spot £30 fine for doing so. You should be able to pick up a copy of the York Cycle Route Map for free from cycle shops, or alternatively you can find PDFs here: (http://www.york.gov.uk/cy...). Bikes are available to rent from a number of locations around the city, including the railway station.
By car
The best advice for driving in York is don't. The roads were designed for carts pulled by oxen, and the city council is actively discouraging car use through a combination of high parking charges and traffic-calming measures. If you are bringing a car to York, your best bet is to leave it in a Park and Ride, at your hotel, or if absolutely necessary, a city centre car park.
By bus
Bus services (http://www.york.gov.uk/tr...) connect all the points of interest in the city but they are not cheap. If more than one person is travelling and the distance is relatively short, a taxi may well be cheaper. However, a one-day bus pass costs £3.70 per person, which is worth considering if you're going to make several journeys in a day.