trains from Bologna to Varenna

You can normally always buy tickets at the station. However buying at the station you pay the walk-up fare, which is the highest.
You can find this out for yourself: Go to www.trenitalia.com and look at the fares: "base" and "ordinario" is what you pay at the station. When purchased in advance you however can get some nice reductions.

However the price difference is not that huge. Looking at 3 October I see that the difference between "Base" and the cheapest discounted fare is about 30,- euro.

30 euro per person is certainly worth saving.

The fares for fast trains - between Bologna and Milan - are those that can be purchased at a discount. You're essentially sacrificing flexibility for price savings and the more you save the less flexible/changeable those tickets will be. The fare for regionale trains - between Milan and Varenna - will always be the same and if you were already in Milan there would be no reason to buy ahead.

If you are confident when you will be up and out and at the Bologna station then it makes sense to save some money and book ahead. But if your plans are still up in the air then you end up paying full price when you get there.

One word of warning the Bologna train station is not like other big city train stations where the tracks are laid next to each other and it has three underground levels and fast trains often leave from the lowest level so give yourself time to get there. Arrive early and if your train track is announced on the board start heading for it immediately because you may have several escalator rides in your future.

Have a great trip - both Bologna and Varenna are interesting but widely different,
=Tod