Traveling to several cities by train. Best way to buy tickets.

You do have to buy each ticket separately and they will be for reserved dates and times whenever you buy them. You're not likely to take unreserved regional trains on these routes. No need to buy until you are ready to commit. Tickets are relatively affordable at full fare, but can be cheaper if purchased in advance (and nonrefundable) at www.trenitalia.com.

Long distance and High Speed train tickets (which is what you are talking about here) are always cheaper if booked in advance. Tickets are for a specific train. When you book, you pick a train, and the tickets say date, train number, coach number and seat number on them. They are only valid for that train.
So yes, you need to book Verona to Bologna, Bologna to Milan, Milan to Verona etc. all separately, and commit to a specific train when you buy each ticket. You will get "print-at-home" tickets sent to you by e-mail, which you will need to bring with you and show on each train.

Note, I am talking about long distance trains, "Regionalle" trains have no discount for advance purchase.
Look up times and book tickets at the Trenitalia website: http://www.trenitalia.com/tcom-en

Be very careful in selecting your trains. Just looking at the Verona-Bologna leg for tomorrow, I see some Freccia+Freccia connections that take 1 hr. 50 min. and cost 46 euros (would be much cheaper if I were pricing for May). There are a couple of Regionale Veloce direct trains that take only about 1 hr. 30. min. and cost less than 11 euros. What you would probably want is one of the Freccia direct trains taking less than an hour and costing 24 euros, or the EuroCity that's 54 minutes and costs 19.40. Like the Freccia fares, EuroCity fares are lower if tickets are purchased far in advance. The Regionale prices do not change as the departure date approaches.

I've seen some odd combinations of price and transit time on fairly short Italian rail trips since using the fast (expensive) trains that don't stop everywhere sometimes means extra mileage. It pays to look at all the options carefully before buying.

Thanks all. In some cases I will be unable to commit to a specific train, so I'll have to pay the higher fare. In others, I will be able to book in advance. Thanks so much!

Thanks all. In some cases I will be unable to commit to a specific train, so I'll have to pay the higher fare. In others, I will be able to book in advance. Thanks so much!