The most efficient route would be Venice > Florence > Cinque Terre > Rome, as that minimizes transportation times between locations. One other thing you'll need to determine is which of the five C.T. towns you'd like to stay in (which will also be important when buying train tickets).
I haven't checked, but as I remember these are the travel times for the routes I listed above (along with the stations you'll be using).....
- Venezia S. Lucia to Firenze SM Novella - about 2H:05M via high speed Freccia or Italo trains.
- Firenze S.M. Novella to ??? C.T. town - about 3 hours, with two or three changes depending on which departure you choose. The trains will be a combination of Regionale, InterCity and Freccia trains operated by Trenitalia.
- ??? C.T. town to Roma Termini - about four hours via Regionale and then InterCity or Freccia trains operated by Trenitalia. You'll first need to get from your C.T. town to La Spezia and then connect with one of the direct (no changes) trains to Roma Termini. There are numerous types of trains on that route, but I'd suggest connecting with one of the direct trains.
You can see all the rail details on the Trenitalia or bahn.de websites. Schedules for next summer won't be posted yet, so just use schedules for the next week or so, using the same day-of-the-week that your trips will be on.
If you haven't travelled by train or other public transit in Italy before, there are several potentially expensive "caveats" to be aware of. If you need further information, post another note or send me a PM.