Hi,
If you do want to look at the interrail Italy passes, they are listed on this site. You have the choice between 3/4/6/8 travel days within 30 days, and for each day when you use a train, you check off one day. With one travel step between each two stops of your trip, I assume you could do it with the 6 day pass (you are probably going back to rome in the end, right?).
There's a major difference in the price between "youth" (before 26th birthday) and adult price, Interrail has always been a thing for young people (I'm feeling old now after writing that
). Eg 6 days under 26 would cost you 103 quid, which is fair enough but the main problem is that quite a lot of major trains in italy have compulsory reservation which on many trains (normal IC/EC) is 5€ but on some up to 25€ (see here - seems to be a good idea to avoid Eurostar, Cisalpino and TBiz trains in that case).
So in the end it's somewhat likely that you come to roughly the same price, or you get slightly cheaper with individual tickets (where btw you also have to pay the reservations, they are sometimes even more than the fare itself, similar to flight ticket price vs. airport charges..), but that's difficult to calculate ahead in detail. If you go individual, you might actually be able to book just one return ticket from Rome to Venice, the other cities are more or less on the route, except Pisa which you may have to ask to go through. Ask how long it is valid and if you can "split" the journey in that case.
For things to see, Rome Florence and Venice are certainly top places and "must see". Pompeii certainly interesting. Pisa not sure if there's much to see (apart from that tower thing), and Bologna i'm not sure if there's that much either, but of course if one is boring you can just move on to the next. One thing I would suggest if you manage to fit it in would be a side trip to Siena from Florence. It's a gorgeous city, very nice but not as big and busy as all the big ones that you visit. I don't have the information on hand how to get there but if I remember correctly it's not much more than one hour bus trip from Florence (possibly there's a train too).
Enjoy your trip!
Cheers Luzian