First time to Italy and Spain

You don't really have enough time for all of that with 13 days...especially since a Rome to Barcelona ferry would take a day. With a 13 day trip, you should choose 3 or 4 cities max. Venice, CT, Rome and then maybe somewhere south of Rome that interests you. Depending on where you fly out of in the US, you could shape your choices by any nonstops from your home city. I've done Italy with Barcelona, just because I really wanted to, but I started in Barcelona, then flew over to Venice, headed south and flew home out of Rome.

There's plenty for you in Italy though...you could easily your 13 days there. Or Spain if you'd rather.

With only 13 days, I would not waste a day traveling from Rome to Barcelona by ferry. Most likely it would be cheaper to take a direct flight from FCO to Barcelona and the trip is less than two hours. The ferries are not cruise ships and accommodations are adequate but not much more. Realize that on a 13 day trip, normally little is accomplished on your first day in country due to jet lag, lack of sleep and checking into your hotel. Similarly, your last day is just a travel day to return home.

Personally, I would spend all your time in Italy to avoid a lot of travel time on planes, trains and ferries and truly enjoy visiting the country. Far niente! You need at least 2 nights in Venice, 1 night (perhaps) on the way to Cinque Terre, 2 nights in CT, 1 night in either Lucca, Pisa or Orvieto, and 3 in Rome. And this is at a minimum, leaving very little time for Spain.

If it were my trip, with only 13 days, I'd choose either Italy or Spain, not both.

But for sure, I would not take a ferry from Rome (Civitavecchia) to Barcelona. I would fly. Let me ask you a question. If you had to go from Miami to Cancun, and you didn't have a whole lot of time, would you fly or take a boat? Miami to Cancun is exactly the same distance as Civitavecchia to Barcelona.

we considered flying into barcelona, the fares are quite cheap if you plan well

You said you want to avoid large cities as much as possible, but of the specific places you listed, all but the Cinque Terre are cities. If you really want to avoid large cities, this will be much easier if you stay in one country. For instance, you could start in Venice for 1-2 days to get over jet lag, then rent a car and see small towns as you meander toward Rome. Return the car in Rome and spend 1-2 nights there before flying home.

Without a car, you could still do this, but it would mean basing for part of your trip in a bigger town with bus or train service to the smaller towns. While some places are well served by train, you will need buses to fill in the gaps, particularly in Tuscany. For instance, you could go by train from Venice to Padova and then Ferrara (both great places), then base in Florence (not a small town, but has the best bus and train service to smaller places in Tuscany). If you don't mind more work with bus schedules, you could stay in Siena, which is smaller than Florence. Then spend a few days in the Cinque Terre, then head to Rome for a few days before flying home.

You could do something similar in Spain. Start by flying into a larger city like Seville, Barcelona, or Madrid, then get to smaller towns in the region by bus or train (again, buses often work better for many places). Look at any guidebook for the most popular small towns near these cities. With 13 days, you can see more than one region - just don't try to see the whole country in this amount of time.

Do heed the following tips, many already given above:
1) The day of arrival is often a jet-lagged haze. Don't plan on any heavy sightseeing (no Prado or Vatican Museums, or you'll be miserable, as I was when I went to the Prado on my first day in Spain in 1995).
2) The day of departure is taken up with leaving, so you usually can't do any sightseeing on this day.
3) Two nights in a place equals one full day. One full night in a place is less than one full day.
4) Changing locations takes about a half day even if the two are close together with fast transit between them (e.g. Florence to Rome). If they are farther apart and/or a flight is involved, you lose much of a day, door to door, and including re-orientation time. This is even more true if you're changing countries. Don't underestimate how disorienting it can be to get used to a new country for the first time.
5) A series of one-nighters is tiring. While one night may be enough time to see a small place, you want to intersperse these with longer stays.

Thank you everyone for great advise so far. We will not take the ferry and will instead either take a flight from Rome or Pisa to Barcelona. I appreciate so much your suggestions and look forward to reading more as they come in.

If you travel on day 1 to Venice, spend that night and the next in Venice.
Train to Cinque Terre (assuming that is a must see for you). I would go to Lake Como instead. Spend 2 nights, or more if you are so inclined.
Train to Rome and spend 2-3 nights.
Fly to Spain, but you didn't say where you wanted to go in Spain. I would avoid Madrid if you want to avoid big cities. I highly recommend Seville for a couple days.
The way I see it, you still have a few days to play with.
Have a great trip.