The easiest way to get to the towns of the Golden Ring is by train. Trains going from Moscow's Yaroslavsky train station will take you to Yaroslavl and Rostov. Trains from Moscow's Kursky train station run to Vladimir several times a day, while trains for Nizhny Novgorod always stop in Vladimir. Unless you speak some Russian you might find it easier to ask a hotel concierge to check the train schedule for you; buying the tickets at a train station is hardly ever a problem even for foreigners.
There are plenty of trains running on the main routes (Moscow–Yaroslavl and Moscow–Nizhny Novgorod) on which all the towns in this region lie, so it's quite easy to travel between towns here, as well as to and from Moscow. Two types of trains will get you to most of these towns: elektrichkas (suburban commuter trains) and normal long-distance trains. In addition to being cheaper, elektrichkas run more frequently. But they are also a bit less comfortable, and there's no reserved seating. Check with a local travel agent in Moscow or at the station itself for train schedules. You can also find the schedules online on the Russian Railways website www.rzd.ru. With the exception of traveling by elektrichka at busy times (Friday evenings and weekends), you shouldn't have trouble getting a ticket the same day you wish to travel.