You will find that due to the size of the town, most of Williams is fairly easy to walk. The exception is around Historic Route 66, where there is enough traffic to pose a significant obstacle to crossing the street, and even there it is possible to get across the street most of the time on foot.
There is a map of the town located near the "Grand Canyon Gateway" arch that represents the "main entrance" to the city on Grand Canyon Avenue, but there is no public parking near the map. If you are staying in either of the hotels near the "Gateway" it is fairly easy to just walk over to the arch.
Most of the downtown streets have sidewalks, though in some areas these sidewalks simply vanish for a short distance without any rhyme or reason as to their presence or absence. For example, the Williams Visitor's Center is right in the core of downtown Williams, and has a lot of pedestrian and street traffic near it, yet the sidewalk is simply gone that would connect it to points west - you have to either walk in the street or through their landscaping to get to the sidewalk that does continue west.
Some of the sidewalks in the residential areas are somewhat decayed, and represent the economic difficulties of trying to maintain anything in a town with only a mild amount of tourist revenue to survive.
While the town is walkable, most of the notable nearby attractions are well outside downtown Williams, and you will most likely want a vehicle of some sort in order to drive to those locations.
There is a trails guide to Williams, but some of the driving distances on the map are in the 30 to 60 mile range, with ony one trail really in any sort of walkable distance from downtown Williams (known as the City of Williams Link Trail #124). This should give you some idea of the distances involved in trying to get to sites within the national forest and Grand Canyon.
We decided not to use the train to get to the Grand Canyon as we would not be coming back to Williams after the Grand Canyon. However, if going in the busy summer months, it would be a great idea to go on the train as parking at the Canyon is very limited, and the ride itself is entertaining and fun. The train station is right across the street from the Red Garter and takes you directly into Grand Canyon Village. There is so much to see and do there that a car is unnecessary.