You will be much better off planning to do a "linear" route, that is, since your are only covering ten days, plan to go just between Milan and Rome and look for places you might want to visit in between. There are plenty. Firenze, which is site-ladened, Siena, which is just cool to be in, any place in the surrounding Tuscany countryside, which everyone agrees is beautiful and fits your 'best sites in Italy' criteria, Orvieto, which also has to be experienced first hand, and finally Rome.
As you might sense, there is a consensus on this board regarding limiting lost time in travel once in country, so to speak. Of all the things you can't "buy" while there, time is the most precious. so planning prior to your departure is huge. You can enjoy all of these places by train, (except Tuscany, which really has to be driven) so buying an Italian railpass before leaving is important. They have them to fit your itinerary, but, again, plan how you will be using it beforehand.
As for racing through train stations, that is why they invented those backpack suitcases. My wife and I are in our fifties and have used them for 25 years, one for each of us, when travelling in Europe. You pack light, with wash and wear clothes, and figure if you don't bring it, you won't need it.