high speed train between Rome and Milan

I don't know what you mean by best but there are many you can choose from two separate train companies: Trenitalia and Italo. You can search all schedules and prices here:
www.trenitalia.com
www.italotreno.it
Choose a date not too far ahead of today. Schedules are shown and tickets are sold online about 4 months in advance.
Trenitalia primarily uses Roma Termini and Milano Centrale stations. Italo uses Roma Tiburtina and Milano Porta Garibaldi stations.

Seems most people use Trenitalia here on the RS forums . . . but, we like Italo. Trenitalia is owned by the government, Italo is private - both have high-speed trains, both easy & comfortable. It's always cheapest to buy 120days out. We get 1st class usually because buying that far out on Italo is only about 2-3Euros more for 1st class. But, we've traveled non-1st class many times & been very comfortable.
Study the www.seat61.com website - lots of great info. Seat61 and the Italo website have photos of the trains & seats, so it's easy to figure out what you want to buy.

Apparently, Italo didn't have quite the number of times and locations and stations at one time - but that has changed. We had no trouble booking exactly what we wanted for our trip in October. The website was very user friendly with photos, we printed out tickets with ease and have them saved to our phones too. We were able to get the same stations on both train sites ( Roma Termini & Roma Tiburtina) so Italo must have opened up many more options. Go with whatever one is best for your itinerary.

Trenitalia is a corporation wholly owned by Ferrovie Dello Stato SpA (SpA is Inc. in Italian), which is a holding company owned by the Italian Government.

To say that Trenitalia is run by the Government may be misleading as neither management not the employees are civil servants. The company in fact is structured like any corporation and the employees are employed under a private employment contract unlike civil servants working in a government agency or department. The fact that the government is the majority shareholder does not change the structure of the corporation or the nature of the employment contract.

I think most people use the term 'government-owned' to speak more to the 'official-ness' of the railway. It's a way to differentiate the legit railway companies from the 'all-similar-sounding' travel agencies (Rail Europe, EuroWay, EuroRails, EuroTrain, etc.) that usually charge much more than going to the actual railway, and don't offer anywhere near the true number of trains and routes.

Italo is a new(ish) private rail company, and it only offers a handful of routes, but the routes they do offer seem to have won many riders over. I'm actually planning on checking them out myself next spring. But Trenitalia offers the most inclusive routing for the Italian train system, so it's where most people go for info and tickets.

jpoquette2407, if you want the best prices, buy your tickets ASAP online. Your best prices are probably long gone, but you can probably still avoid paying full price.

As previously referred to, The Man in Seat 61 is a great resource for all things train in Europe. Here you can find info about the trains you're interested in - photos and videos showing what each class of service looks like on the trains you're interested in, seating charts for each type of trains (so you can book seats across from one another or next to one other when the seat numbers don't make sense, know where the snack/restaurant car is, etc), and he even speaks a little to the differences between the Italian train cars used by the companies Trenitalia and Italo.

jp,

The "best" train is usually the fastest (IMO), which means using one of the high speed trains. These are also the most expensive but buying advance tickets can save considerably. Note that the cheapest advance tickets are non-refundable and non-changeable. The high speed trains have compulsory seat reservations, which will be provided with the tickets.

Both Trenitalia and Italo Treno operate high speed trains from Rome to Milan. However one important point to mention is that in some cases these use different stations. Trenitalia uses Roma Termini and Milano Centrale, so that's usually the easiest option. Italo uses mainly Roma Tiburtina and Milano Porta Garibaldi (among others).

At least some of Trenitalia's Rome-Milan service is operated using their very newest trains, the Frecciarossa 1000, introduced in June, 2015.