Rail Pass

Rail passes don’t make much sense in Italy even for people who take several train journeys on a given trip. In your case you will probably save money by purchasing individual tickets online ahead of time at the discounted fares (Super Economy or Economy).

Also read this in RS’ website:
https://www.ricksteves.com/travel-tips/transportation/trains/italy-rail-passes

Then go to www.trenitalia.com (the State railway operator) or
www.italotreno.it (a private competitor which operates only high speed trains).

Check the cost of your individual train journey to your stated cities (check both at full price or discounted tickets) then compare with the cost of passes (see the Trenitalia specific pass below)
https://www.trenitalia.com/en/offers/trenitalia-pass.html

Discounted tickets go on sale up to 6 months prior but sell out as you get closer to the time of travel, but sometime are still available even a week or so prior. Obviously the earlier the better to ensure availability. Full price tickets are available even minutes prior to departure. Trains are rarely if ever totally sold out. Your destinations are along lines where trains are super frequent, even several trains per hour.

Forgot to mention that the Trenitalia website requires you to enter name of the station in Italian. Therefore these are the main stations for your cities. I’m not mentioning the others (Milan has 23, but you will likely use one only).

Venice: VENEZIA SANTA LUCIA (abbreviated in S.LUCIA)
Milan= MILANO CENTRALE
Bologna: BOLOGNA CENTRALE
Genoa has two main stations= GENOVA PIAZZA PRINCIPE and GENOVA BRIGNOLE. Trains stop in both, so it depends where you need to go.
Your destinations didn’t mention Florence, but just in case you decide to visit my city, this is the main station in Florence.
Florence= FIRENZE SANTA MARIA NOVELLA (generally abbreviated S.M.NOVELLA)

When you go to the website enter from the ORIGIN STATION to FINAL DESTINATION STATION.
Then time and date. If your date is far in the future the timetable may not be uploaded yet, so use a date before June.
Once you get an idea of the price for each segment, then compare to the Trenitalia Pass.
Be aware there are also resellers (travel agencies like Italiarail) that sell you bot passes and also tickets. You will generally pay more for those. So use the websites above.

For example the full fare (not discounted) on the fastest train (Frecciarossa) from Bologna to Milan is 48€, but the Supereconomy fare, if purchased in advance, is only 22€. The same fast train from Bologna to Venice is 36€ with the full price, but only24€ with a Supereconomy fare. Be aware that between adjacent regions there are also slower Regional trains and Regionali Veloce trains that are even much cheaper and given the short distances don’t take much longer. Your destinations are journeys are really short, therefore inexpensive, I doubt the pass would make sense for you.

Another note. To go from Bologna to Genoa you need to go though Milan, so it might make sense to visit Milan after Bologna, and then Genoa after Milan.