Train travel in Italy, buy from home or when I arrive there?

Easiest is to use the Trenitalia app. You can use that for all your tickets. Or use Trainline for long distance routes as it will also show you trains operated by Italo, the other main operator.

For long distance services tickets are cheaper when you buy in advance. For regional services however you can always buy tickets at the station, and the price is always the same.

Omio is a reseller, no reason to pay more.

If you can afford the walk-up fares of high speed trains, you can get all train tickets up to 15 minutes before the departure time. On the other hand, If you want to save some money (and your plans are set) you must get the discounted tickets in advance.

Out of curiosity, do you know that your Florence-Milan high speed train will stop in Bologna?
And that the fastest trains from Bologna to Desenzano del Garda-Sirmione take around two hours and one change? The direct train from Milan to Desenzano takes 50 minutes. In short, is there a reason for backtracking twice?

For clarification, we are taking a Rick Steves tour of Tuscany which begins in Florence and ends in Lucca so we will not be traveling from Florence to Milan or vice versa via train. All our train tickets to purchase as described in the original posting are for independent travel which will begin in Lucca after arriving there by bus with RS. Once we leave Florence by bus in the tour, we are not returning there. Thanks for pointing that out but it would not apply in this scenario.

On reading your reply again, I think you are saying it would be easiest to take the train from Milan to Desenzano for a shorter train ride. But will the train from Desenzano to Bologna be 2 hours plus a change anyway? I do appreciate your input and helping me to work this out.

For more clarification needed for myself, will all these routes I described be described as regional, therefore the same price at the ticket counter or are they considered long distance routes?

I know it sounds clichéd, but Did you check where these cities are on a map? In your first message You wrote:

I will need train tickets from Lucca to Milan

And going from Lucca to Milan in less than 4 hours implies taking a train from Florence to Milan. A train that will stop in Bologna.

So, unless you want to backtrack for some reason, you could visit Bologna after Lucca and before Milan. Then you could go to Desenzano in 50 minutes instead of two hours,

or are they considered long distance routes?

It's not about distance, but speed.

Any train called Regionale is a slow train with no reserved seating and no discounts for advance purchases.

Any train called Italotreno, Freccia-something or IC has only reserved seating and you can save some money getting tickets in advance.

Note that discounted tickets can't be purchased on the day of travel and that they come with restrictions about changes and refunds.

Many thanks.