Avoid the Queues of cars and take the Quiet net.
This is a network of routes that have tolerable levels of motorised traffic. It works by linking residential roads together using existing paths. The technical name for this is filtered permeability, since such routes are impermeable to large vehicles, but permeable to cyclists and pedestrians. Since the routes are not greatly useful to motorists, they are relatively car free.
In cyclophobic Britain, many of the routes here are forbidden to cyclists, but the inspiration from this came from cycling in Germany, where every path is shared use unless there's a really good reason for it not to be. Contrary to what you might expect from the letters pages in certain British newspapers, pedestrians are not being killed and injured by reckless cyclists using footpaths, but what this arrangement does achieve is making cycling feel acceptably safe to a significant proportion of the population. It also makes cycling practical, because there's a network of routes. Of course Germany does have far more than this, but making a network like this legal would represent an easy start for the city council. The Dutch also link residential roads with paths, but they never expect pedestrians and cyclists to share the same path.
The Q-Net map is being transferred from Google Maps to the mapping system on this website.