Barcelona to Rome and back

You are really asking for a trip planner and that is hard for us to do. My first recommendations is to consider an open jaw or multi-city ticket. Into Barcelona and home from Rome or reverse. You waste a lot of time and money back tracking. With 16 days you are somewhat limited. Every time you change location you lose at least a half day if not more. Since a lot of your comments about locations to visit are in Italy and the Aips. I might consider flying into Switzerland and head south to Rome via train. Or even into Milan. Barcelona to Rome is a long way via land and to do a circle does not make much sense. And for your fist trip avoid the car. It is a hassle you don't want especially when it comes to parking and the dreaded no travel zones.

You need to spend sometime with guidebooks, a map of Europe so that you understand distances involved and especially travel videos to give you a sense of what is important to you. Your local library is a good place to start.

There are still travel agents sounds like you may need one. Are you planning on renting a car & crossing borders? People on this forum can not plan all your details for you. It works best if you state a plan and ask for insight. What you will save in airfare you will pay in rental cars. You do not need or want a car in Rome.
Good luck in your planning.

Please consider open jaw first of all. I bet in the end you will save the difference. We do it that way all the time. We have traveled with family…adult kids, grandkids. Try not to be on the move every day. Kids need to play at playgrounds. Laundry will need done. Research carefully your stops. Trains are a great way to travel. We dont rent cars in Europe…but many on this forum do. There are good websites to help with train travel. March will give you chance of rain or nasty weather…esp. Alpine regions. You have alot to decide in 6 mnths! Then reservations for flights, lodging and often tours/museums need reservations well ahead. What do those you will travel with suggest or prefer? Find a travel agent. Rick Steve’s guide books and videos at library. Seek out those you know that have traveled these places. Its hard to give more advice until you can get some more solid decisions made. Its a lot to plan and your time is getting limited if you go in March of 2026! Not trying to discourage…but being realistic as a traveler. Best wishes!

I think you will probably eat up most of your flight savings with your transportaton back and forth from Rome. I would very much recommend thqt you decide where. you want your trip and try to find reasonable flights close to those locations. Particularly if you are travelling with young children!
It sounds like most of your interests are in Italy. There are many international airports there, Milan, Florence Venice, Rome and Naples. There must be some flights similarly priced. With 16 days and young children I would focus on one country or region, either Spain with Barcelona flights or Italy. It is too early to train through the swiss alps and I'm not sure where you will drop your car there anyways.