The easy option is to give 5 days to the city and 5 days elsewhere. For ease of arrival / familiarity (and potentially warmer weather outside the city) you might do Rome first and then venture out, perhaps with a car. That's slightly against my usual preference for having a rural place to get over the flight / jet lag first, before hitting a big city.
If you don't mind another city (and you've mentioned you like them), then there's probably not a better place than Rome to connect with, as not only do all roads lead to Rome, but so do most of the train lines.
Frecciarossa, Frecciargento, Frecciabianca , daily connections at high speed at high frequency between the main cities. Find out routes, fares and timetable.
www.trenitalia.com
To get a feel for options, click on the Frecciargento, Frecciabianca (this is a slightly slower service) and Frecciarossa 'more information', which has maps showing the connections for these superb fast & comfortable train services. Note that Italo now provides high speed competition as well, so worth a look.
Some thoughts
- Florence or Naples offer what might feel like opposite ends of the spectrum. The former fancy and strongly touristed, with lots to see, the latter more intimate & soulful / real / gritty, but also with plenty to see.
- Bologna a city that has tourism, but is far from beholden to it. Arguably the best food shopping in Italy, and with the option to visit producers of Balsamic Vinegar, Parmesan & Prosciutto. It's also got lots of porticoed streets, including plenty of genuinely historic ones, so the weather isn't a limiting factor
- La Spezia acts as a gateway to Cinque Terre, and Beerman has posted a lovely trip report with lots of photos to tempt you.
All of these can be good for airports with good coverage, so you can fly into Rome, but out of another airport like Bologna, Pisa or Florence.
So many other places to consider as well, but I'd definitely applaud taking a simplistic transfer option of a direct fast and comfortable train, which itself can be a highlight (especially if you take along a nice picnic)
