I want to fly in to Venice, stay 3 days and take a train to Cinque terre. However, I don't see a very conventient way to get there by train.
Carlene, Where are you trying to find information on "convenient" ways to get there by train? Checking the Trenitalia website, there are LOTS of trains each day. The trip will be 5 - 6 hours, with at least one change. Are you more interested in the fastest travel time or fewest changes? My preference would be a train departing Venezia S. Lucia at 10:32, arriving Monterosso at 15:55 (time 5H:13M, two changes at Firenze and Sarzana, reservations compulsory on the first leg). You didn't say which of the five C.T. towns you were travelling to, so I used Monterosso for searching. Happy travels!
Not sure on your definition of "convenient" but it's 5.5hr-7.0hr train journey to get from Venice to the CT. No way around it. Ken describes one of the fastest run of the day.