There are a few daily direct trains from Genova to Rome : 5-6h journey. Check timetables on trenitalia.com.

Otherwise other alternatives include taking a train to Milan to pick up the high-speed line to Rome. A detour and you miss the Cinque Terre along the way but an option nonetheless.

Are you considering visiting those places as day trips from Rome ? Apart from Florence I'd very much be against that. All those cities are quite far away from Rome and as daylight is still quite limited in February you won't have much time there.

Of course if you plan to spend a few nights in those cities all good.

About tickets : I don't think the Eurail Pass will be good value as you need to add a 13€ reservation on any Frecciarossa high-speed train while advance tickets are quite cheap. Check fares on trenitalia.com and italotreno.it

Questions welcome

Thank you so much for you response.

Do you perhaps have any suggestions on what we can do in the other cities?

Do you also know of affordable accommodations

Thanks

There are plenty of things to do !

Venice : just walk around, maybe visit a Palazzo or two

Florence : walk around too, Galleria degli Uffizi (world class museum), Campanile, cathedral, sunset from one of the hills surrounding the city... 

Cannot help on accommodation but have a look at what Booking and Airbnb offer.

Thank you so much for your guidance I appreciate it.