Three of us will be traveling from Rome airport to Rome Termini, then to Florence, then to Civitavecchia, then to Venice, then to back to Rome and finally back to airport between May 26 and June 13. Should I make train reservations? Do they need to be first class?

Hi Donna, I did almost the same trip last year. We flew into Rome, took the Leonardo Express (33 euro) to Roma termini. Then from Roma termini we took a train to Florence. From Florence ( Santa Maria novella) you can get anywhere. 
as far as 1st class or 2nd class, we have a Eurail pass and bought 2nd class. The seats are only 10-euro each way. The Eurail pass is the best deal for the money. Unlimited train travel within 3pm days depending on how many days you purchase. We bought 8 days for 443.00 euro with senior discount. I hope this helps and enjoy your trip.

NONE of the tickets need be 1st class, and I'm not sure if there is a 1st class on all these legs.  Certainly the 1st class cabin is nicer for the longer trips.  Depends how much you want to spend for the duration on the train.  It appears your longest trip is Venice-Rome which is 300 miles, 3.5 hours on the freccia.

Yes, make reservations for the big journeys - 1st or 2nd class tickets with seat assignments.

I think all necessary homework can be accomplished on web sites (ww.TrenItalia.com50 or Eurail.com32) all in English.  You should be able to book all the big parts of trip and print tickets from the comfort of your own home.  For the littler trips, there will be self-service machines in lobbies and/or ticket agents behind the glass (they don't all speak much English, maybe enough to sell a ticket and wave you in the direction of correct "binario")

In your research, consider if a RAIL PASS (rather than purchasing trips as individual tickets) is a lower cost alternative.  You still book specific reservations for certain legs - see how  it works when doing the homework.

Happy travels!