From Vernazza any of the five Cinque Terre towns (or more than one) would make a nice day trip.  The five towns:  Vernazza, Riomaggiore, Manarola, Corniglia, and Monterosso al Mare all have different vibes and things to see and do and are all less than an hour apart by train.  The two closest towns to Vernazza are Monterosso and Corniglia, both of which people often choose to hike to rather than taking the train.  (Hikers will need to buy a hiking pass before going, so be sure to ask at your hotel.) 

Monterosso is a larger town than Vernazza with more traffic and tourists.  It has a nice beach as well as an old town, dating back hundreds of years.  The old town is full of shops and cafes and is fun to walk through.  Corniglia is the only one of the five towns not located on the coast.  It is known for its wine making and is home to several vineyards.  Visitors to the town will find many wine shops, restaurants, and less expensive hotel rooms than in the other towns. 

For more information on Cinque Terre's towns visit the Cinque Terrre website.