Leave your car in Levanto and take the train to Monterosso (attention: you should look for departures on the web because sometimes they only depart just once every hour). That's the first beautiful town you should see and start your trip. You can walk along the high coast to the other places and it's really worth it, but you have to be sporty.
Some parts of the trail are currently closed where you have to use the train again to get to the next village. All those villages in the Cinque Terre are connected by a train route, so that's probably a good idea to walk just one direction and get back where you started by train.
If possible you should plan your visit outside the usual holiday season, because there is very much "traffic" on the trail between the villages and they are sometimes pretty narrow.