Train Tickets - buy ahead of time, or buy there?

All of the places you mentioned are served by regional trains. In Italy, regional trains cost the same booked at the last minute as booked ahead of time, so there's no benefit to booking early. (They can't sell out because there are no reserved seats, but best to get to the station early to get a seat - sometimes they get crowded so you have to stand. I have.)

That said, there are some IC trains that run too on some of these routes - these trains are reserved and cost a little more (you might save a LITTLE booking ahead on these trains, perhaps not much compared to some of the other express trains in Italy). As long as you stick to regional trains, no worries about booking ahead, at all.

If you miss a regional train, you have up to four hours after the original departure to use your ticket. This is not the case for other trains with reserved seats.

Also, you MUST validate any paper tickets you buy for regional trains in the yellow boxes at the stations!!! Just slide your ticket in to the yellow box and get a stamp. Watch the locals do it. If you board a regional train in Italy without a validated ticket, you risk a steep fine. That said, if the ticket you buy is bought online for a specific time, no need to validate it. (Or if you buy on your phone with say the Trenit app, just show the ticket on your phone, no paper to validate.) But if you buy a paper ticket at a train station from an agent or from a ticket machine, you MUST validate it, even if you bought it just a second ago.

You can certainly buy your train tickets "on the fly" when you get to Italy.

The only tickets you may want to book in advance are the long distance, high speed trains (freccia), that you may need to take, say from Florence to Rome, in order to save some money.

Most of the trains you'll be taking for trips (Genoa to Vernazza, Vernazza to Lucca, Lucca to Pisa, Pisa to Florence) will involve Regionale Trains, so the price remains the same whether you buy in advance or the day of travel.
Regionale train tickets are valid for 4 hours from the time they are validated ( date/time stamped at machines near platforms), so if you miss one train, you can take the next one.

However, you cannot do this with freccia train ticket, as these are like airplane tickets, and you have a reserved time, date and seat.

Here's a very helpful website you may want to read: seat61.com/Italy-trains

I like to buy my Regionale train tickets on trenitalia.com on my smartphone, and that way I don't have to wait in a possibly long line at the train station, or at the self-serve ticket machines.

Enjoy your time in beautiful Italy!

As you want to maintain flexibility, buying tickets in Italy is the best solution. As others have said, many of the trains you'll be using will be Regionale and there are no discounts for buying those in advance. Again as previously mentioned, it's important to validate (time & date stamp) locally purchased Regionale tickets! If you're caught with unvalidated tickets, you'll be subject to hefty fines which will be collected on the spot.

While you could save a few Euro with advance purchase tickets for the express trains (Freccia or Intercity), the routes you'll be travelling are all relatively short so the savings wouldn't be that much. The tickets for those don't have to be validated as they're specific to train, date and departure time. It's important to know which type of train you're boarding, as if you board the wrong express train, you'll again be subject to hefty fines!

I highly recommend the seat61 website mentioned above. It contains a TON of useful info!

We were in Florence last month and did a day trip to Pisa, buying tickets on day of travel from a machine, in English, with cash, very easy to do. We also did day trip to Venice from Florence, buying tickets one day in advance from an agent at the Florence ticket counter, and we got reserved seats next to each other even though it was pretty last minute on a very busy route. Summer will probably be a bit more busy, but I bet if you buy a couple of days in advance in Italy you would still be ok.

In addition to the man in seat 61 website, which I agree is outstanding, I recommend loco2, a British website that sells train tickets. It's easy to use the loco2 website if you want to purchase tickets, but I also found loco2 very useful for two other things: (1) schedules (you can figure out what times trains will run from Florence to Lucca, Florence to Pisa and back, etc., before you leave the US, and (2) you can estimate what it's going to cost you if you buy at the last minute (use the loco2 drop down menu of ticket options and use the high price one to get a worst-case price scenario if you buy on the day of travel).