mary,
So, if I understand correctly, I will be using mostly premium trains.
Not necessarily. Many routes have both premium trains (which is my term) as well as Regionale trains, so which one you'll be using depends on which ones you decide to use. You may be using both types on some trips.
Is this Trenitalia?
You'll most likely be travelling via Trenitalia in most cases, but difficult to say without a list of which routes you plan on travelling. Italo may be a possiblity for some routes, but I'd suggest "keeping it simple" and just stick with Trenitalia.
Where do the regional trains go?
The Regionale trains go to the same locations as the premium trains, and some places the faster trains don't go.
Or is it just that these are the slower trains?
Yes, they are slower, especially as they make many stops and don't travel at high speed. They're somewhat the equivalent of a "bus on rails".
Can I get reservations on the premium trains the day I want to travel? Just go to the station and buy a ticket for (hopefully) the next train?
In many cases you can buy a ticket on the day of travel for the next train, but there are exceptions. If it's a holiday or other occasion when a lot of people are travelling, all seats may be sold-out. As each passenger has an assigned car and seat, when all the reservations are gone, anyone else is out of luck. AFAIK, there's no "SRO" on premium trains. I generally prefer to buy tickets the day prior to travel, as that avoids the problem of arriving at the station and finding huge queues at both the ticket office and Kiosks, which could possibly delay my departure.
From what I understand, trains run frequently -- every half-hour or so -- so if we arrive at the station 30 min before we want to leave, worst case scenario would be maybe waiting an hour?
Not necessarily. Trains on some routes run frequently, but that's not the case on all routes. You can check the TRENITALIA website to determine the frequency and departure times of trains you're planning to use.
Do reserved tickets have to be validated?
No, tickets with compulsory seat reservations do not have to be validated, since they're only valid for ONE specific train, date and departure time.
Is it better to buy round trip tickets for the day trips -- or can we just purchase return tickets when we are ready to return to our base?
In most cases, I prefer to buy round trip tickets for day trips, as that means I don't have to waste time buying tickets for the return trip. However, if the journey will be made on a "premium" train you MUST be sure to use ONLY the train specified on the ticket for the return trip. If the journey will be via Regionale train, you MUST validate the ticket prior to boarding the train. DO NOT validate both tickets at the same time, as they have a "shelf life" once validated (usually about 6 hours).
If luggage is put on racks at the end of a car, is it checked somehow or just left there? I'm not sure I'm comfortable just leaving it.
No, luggage is not "checked". Passengers just find an empty spot on the rack and drop the bag there. If you place bags on the rack, watch them closely. At times the racks can be full, which means that some luggage is left on the floor, somewhat protruding into the aisle. With luggage it's often necessary to adjust to conditions at the time. If the train you're using has compartments, it's sometimes possible to keep larger luggage items with you in the compartment (with smaller carry-on size items on the rack above your seat in the compartment). If the train doesn't have compartments, it's sometimes possible to place larger items in the "A" between the seat backs. You'll have to "play it by ear".