I have a few comments to add to the others.....
do we need to make reservations in advance?
You'll only need reservations for travel on the "premium" trains such as the Freccia high speed versions. You don't "need" to make reservations in advance, although at busy travel times (ie: holidays) if can be worthwhile to do so. If you buy P-P tickets for those trains, reservations will provided with the tickets. Note that the reservations will be specific to a particular train, date and departure time, and can ONLY be used on the ONE train specified on the ticket. If you board the wrong train, you'll be deemed to be travelling without reservations and will be subject to a hefty fine, which will be collected on the spot.
Reservations are not possible on Regionale trains so it's just as easy to buy tickets for those at the stations, either from a Kiosk or from a staffed ticket office. NOTE that Regionale tickets MUST be validated prior to boarding the train on the day of travel, or again you'll be subject to hefty fines!
is there any great benefit in purchasing first class tickets on the train (if available)?
I suppose that's a matter of opinion. I don't believe you can purchase tickets on the train. You'll likely be fined if you try that. I've never felt that first class was worth the extra money, as second class is perfectly comfortable. However, one exception - if you find that all the second class tickets for a particular train are sold-out and that's the train you want to take, I'd consider buying first class tickets as those are less likely to be sold out.
Our travel days are set but I would rather not have to decide before we leave what time we want to depart to the next stop.
In that case, I'd suggest buying your tickets at the stations in Italy, perhaps on the day prior to travel. I like to do that so that I don't have to wait in queues or whatever on the day of travel. If I have my ticket in hand, I can proceed directly to the train. When buying your tickets at stations just before each trip, you'll likely be paying "Base Fare" (the highest level), but that provides the most flexibility in travel times. Especially with Regionale tickets, you can buy those on the day of travel as there are no reservations and they never sell out.
Oh, one last question....Is there a cost savings to buying your train tickets ahead of time?
The answer to that is both "Yes" and "No". There's no cost savings for buying Regionale tickets ahead of time, and it's difficult to do so. There can be a cost savings for pre-purchase of tickets on the "premium" trains. As Frank mentioned there are three classes of tickets - Super Economy, Economy and Base. The Super Economy tickets can be as low as about €9. However, as you might appreciate those usually sell out fairly quickly and once they're gone the next level is Economy, which is a mid-priced ticket. One important point to note with the premium tickets is that once purchased, the travel date and time is "set in stone" so you'd have to be very sure you could commit to travelling on the date and time you choose.