Train advice

Advance buying is only relevant for the high-speed Frecce and Intercity trains. Not so much because the train can be sold out, but for the often considerable discounts which come with advance buying.
The other trains - Regionale and Regionale Veloce - have no seat reservation and so cannot sell out, and have a fixed price.

For high speed trains, they work just like airline tickets. There is nothing to say you can't just go to the airport and walk up to the desk, slap down your credit card and say "Put me on the next flight to X." You will be paying the highest possible price. Same for trains, it is just not as big a number. Ex. Venice to Florence is about 57 EUR 2nd class base price. They are unlikely to be sold out, and trains go about once per hour. And there are two different train companies operating that route, Trenitalia and Italotreno.

Buy in advance to save money, but you do lose flexibility. What is that flexibility worth to you?

Here is an example of the savings with tickets purchased ahead of time, using Italo trains from Florence to Rome:

Tickets bought today for travel tomorrow range from 48-56 euros per person.

Tickets bought online today for travel in late October (10 weeks from now) are as low as 21,90 euros per person.

Trenitalia prices will be similar. And for longer distances, the savings are even greater. Note that these savings are applicable to the high-speed (Frecce and InterCity) trains, not regional trains (like the short-distance trains serving Cinque Terre from La Spezia).

Finally,

As the previous reply mentioned, whether to buy tickets in advance will depend on which type of train you'll be using, and also whether saving money is a higher priority than spontaneous travel.

With Regionale trains, there's no point to buying advance tickets at the price is always the same. While those tickets can be purchased in advance, it's just as easy to buy them at local stations either from a staffed ticket window or an automated Kiosk. One important point to mention though..... you MUST validate (time & date stamp) the tickets prior to boarding the train on the day of travel. or you'll risk hefty fines which will be collected on the spot!

With the express trains (Freccia, Intercity), you can save money on those with advance purchase tickets, however you must be willing to commit to a specific departure as the tickets for those trains include compulsory seat reservations which are specific to train, date and departure time. If you board any other train than the one listed on the ticket, you'll be deemed to be travelling without a valid ticket and again, hefty fines collected on the spot! If you buy Super Economy tickets well in advance, those provide the greatest discount, but those are non-changeable and non-refundable after purchase. When the quota for those is sold out, the pricing moves up to Economy tickets, which are not as cheap but have slightly more flexible rules. The highest price tier (and the one you'll pay for walk-up tickets is Base Fare.

You can see all the train details (type of train, schedules, prices) on the Trenitalia website.

In addition to trains operated by Trenitalia, you can also use Italo Treno on the high speed routes. AFAIK, those have the same rules as those for the Trenitalia express trains.

So, the decision is yours..... save money or spend more for spontaneous travel.

Edit - I was posting at the same time as the previous two replies, so there may be a bit of duplicate information.

Two additional thoughts:
For major train rides on Freccia etc. trains, I will often buy online far in advance, to save money yes, but also to give my schedule some “anchor points” to plan around. Give up flexibility but gain savings and structure. Just like air tickets.
Also, I would much rather have my ticket in hand before I arrive at the RR station to travel. That is even for regional trains, which I try to buy the day before I travel. That is because various times I have arrived just in time to travel, but delays in buying a ticket (ticket machine did not work for me, or long, slow lines at the machines or ticket counters) have caused me to not make my train, needing to travel later, maybe slower. With ticket in hand, you just arrive, find your track, and get in.

Following up on Larry's advice above, consider downloading the trenitalia app to your smart phone. With it you can search for and book both the frecce's and the regionales as far in advance as you're comfortable with. Your tickets are then sent to you as attachments to a confirmatory email - no need to mess with paper tickets, plus they're already validated so no need to worry about validating them before departure.
Thing is - since the regionale tickets are already time stamped with the specific date and time of the train, they're not quite as flexible as the paper tickets for which you have a 4 hour window on any train heading in your direction after you're validated it in the machine trackside. That's the tradeoff.
We found the app to be very helpful during our own recent trip to Italy.

Italo often has discounted fares available after Trenitalia's discounted fares are sold out (similar restrictions), so consider that option if you want to save money but prefer to lock in a specific train closer to your travel date.

I strongly recommend not buying your train tickets for your day of arrival (if you plan to take a train on the day you arrive).....Even though you'll have to pay the higher base price, it will save any worry you might have if your plane is late or there are longer lines than you anticipated.

I've bought my Freccia tickets already for my October trip, with the exception of my first day - I plan to arrive in Rome and immediately take a train to Padua. I'm not going to buy my ticket until I am actually in the station, because, while I hope I'll be able to catch the 12:03 train, I might need to take the 1:50 or 2:50 instead.

Speaking of Trenitalia - when you use their website, you have to use the Italian place names (even if you're using their English page) - so use Roma, Padova, Firenze, or Napoli instead of Rome, Padua, Florence, or Naples.

What Larry said about those long lines!! It's happened to me too. Even if it means stopping by the station the afternoon or late evening before, it's worth having the ticket in hand.

There is no need to go to the departure station to buy tickets 24 hours in advance. On top of other stations, travel agencies and the internet, many Tobacconists sell regionale trains tickets, too.

I may add that there is no need to stand in line to get tickets just before the departure, either. Trenitalia's app allows purchases from 180 days in advance up to 15 minutes before the scheduled departure time.

Thank you all so much! I appreciate your help. Half the price may be worth it! I will check out the schedules! Thanks again!