Especially in high season, the 5 Terre Expres train is the most comfortable, fastest and cheapest way to get around: the railway line connects the five villages (Riomaggiore, Manarola, Vernazza, Corniglia, Monterosso) with frequent hourly rides and it takes about 4 minutes from a town to the other. Stations are inside the villages so you will not need other transport (bus or taxi) except for Corniglia that you can reach only by bus or on foot through a long uphill road.
Tickets can be purchased directly in the stations. From April 1st to November, the ticket for each ride (you can buy tickets at each station), costs 4 Euros per adult and 2 Eros for children from 4 to 12 years.
If you plan to take more than four trains on the same day, the best option is the Cinque Terre Card, (you can buy it at each station) which includes all the trains between Levanto and La Spezia for the whole day, free wi-fi, free entrance to the Blue Trail and much more. If you want to know the rates and the services included, you can visit the website of the Cinque Terre National Park dedicated to the Cinque Terre Card.
Train schedules change frequently, according to the period. Here are the times currently in force. Anyway you can be always updated, visiting the website of 5 Terre Express.