ttrip to Italy

How many nights exactly did you book in Rome?

I only booked one night in rome

Shanika, with only 7 days, that doesn't give you much time to see each city that you want to. Which 2 or 3 (including the one you fly into and the one you fly out of) do you want to visit most? Where are you landing & leaving from? The big cities, except for Venice, all have hop on/hop off buses. You will be so stretched for time, that that's about all that you will have time to do... unfortunately.

I fly in and out of bologna. My top two cities would be venice and rome.

just do rome and venice imo. you will have great memories. do florence another trip through tuscany

In that case, divide your nights between Rome and Venice. With your first and last nights in Bologna. From Bologna, take the fastest train service in the morning to Venice, spend 2 nights. That should give you at least 1 1/2 days of sightseeing in Venice. Then, again, take the fastest direct train from Venice to Rome in the morning and spend 3 nights in Rome (2 1/2 days of sightseeing). Can you add 2 nights to your existing booking in Rome (same hotel)? Depending on when that night is, you may need to flip Venice/Rome to Rome/Venice. This plan assumes that you have 7 nights: 1 + 1 in Bologna, 3 in Rome, 2 in Venice.
That would give you time to actually visit stuff too, like the Vatican, the Colosseum, etc.. You will even get a chance to sit down and enjoy a coffee or gelato, or even both :-)

As to what to do in each city, I really can recommend the Hop On/Hop Off (HO/HO) bus for Bologna on the day you arrive. Make sure that you walk around quite a bit that day too, it'll help get over the jetlag. Rome also has a great HO/HO bus that you can take over a couple of days. As to sights, go to the nearest bookstore and read the guidebook sections for these 3 cities. Buy the one guidebook you like the best. Then "A" and "B" list the stuff you most want to see. Make a tentative plan for each of your days, "clustering" these things in the same neighbourhoods & HO/HO stops, so that you don't waste time zigzagging all over town. Come back and ask more questions as your plans progress.

I kinda feel a little different about that. Strictly my opinion, but would it make more sense to travel south from Bologna to Florence, spend a couple days, then further south still to see Rome, stay a couple days and then go from Rome to Bologna? If you go from Bologna to Venice, you travel north, then you are backtracking past Bologna to go to Rome (and the same is true even if you go to Rome first). Fast train or not, that's wasting time going in opposite directions. Since it's obvious you can't do all 3 big dogs (Rome, Venice and Florence) this trip, I would opt to sacrifice Venice just for the sake of being practical. Once again, just my opinion and either way, you're seeing the beauties of Italy.

According to the Trenitalia website, these are the fastest traveling times I saw, assuming morning departures: Bologna to Venice: 1h20 Bologna to Florence: 0h40 Venice to Rome: 3h45
Florence to Rome: 1h30 Rome to Bologna: 1h30 The time differences aren't big enough, IMO, to sacrifice first choices - especially since all of these are direct trains. So, ultimately choose what you want to see & do most... BTW, if you choose to give 2 nights only to Rome and squeeze the 4th city in one night - which you may still be considering, I really think that it's squeezing in way too much, and would advise against it. You will be seeing a lot of blurry countryside and lots of interesting stuff as you whizz by, without being able to visit & actually see. I think that the head-scratcher for me is the Bologna flights.. as for what you really wanted to do, it's more common to fly into Venice or Rome, out the other, and put Florence between the two...