I'll try to expand.
When you arrive in Amsterdam, you walk out of the jetbridge into the international terminal at a gate, similar to a domestic flight, which is unusual for many locations. Look for signs for connecting flights and proceed that way. Somewhere along the line, there will be monitors, that depending on your departure gate, will direct you to different immigration lines. Once you reach the immigration line, there will be monitors that list tight connections flights, if your flight is on there, then you can go to a separate line for fast service. However, I have had as little as an hour connection, and that flight is not on the monitor, and I made my boarding with 20 or 30 minutes to spare.
As others said, if you review your itinerary, it will give you a Delta number for the Amsterdam to Rome leg, but it is very likely operated by KLM, but still no problem.
Once landing in Rome, the typical traveler will take the Leonardo Express to Termini Station, then the first train to Orvieto. Maybe download the Trenitalia App, and if you have service, you can search for the best train option. Your choices on trains are either the RV trains, fast regional trains, or Intercity train. You can go to a manned window, buy a ticket from a machine, or buy from the app. Each has their own oddities, just that if you buy paper tickets, validate them, electronic tickets, check in, unless you have reserved seats (though not possible on the Leonardo express and regional trains)
Private driver is fine to Orvieto, but way to rich for my blood. But then I never understand the people who are zombies once they land, I usually have enough adrenaline, even now after a couple dozen trips, to keep me going.