While your question is perfectly reasonable, unfortunately it has no definitive answer. That's because Rome, Venice, and even Milan have much more than you can see in your limited time, and no two people will have the same "must see" and "must miss" lists.
It's a similar situation with the way you allotted your time in each place. Some will say skip Milan and/or Venice, others will say cut the Cinque Terre to allow more time for Venice. And while many are not fans of Milan, I am, and I think a day there can be very worthwhile. One advantage of Milan is that it has a very efficient Metro, so you can connect sights easily and quickly.
A day in Venice is certainly rushed, but on the other hand if you stay out of "sights" and just treat Venice itself as the sight, I think you can have a great day. In your situation, I would not plan to see anything indoors (unless you had a special interest), and would just hit the back streets and take vaporetti around. This will also further your goal of saving money.
However, I do agree that landing and staying in Milan one night, then going to Venice and staying one night, then going to the Cinque Terre (with a long train ride) by the third night, is going to be exhausting. Therefore, and for that reason alone, I'd pick either Milan or Venice for 2 nights. And I wouldn't plan to stop in Florence (unless you cut something else), as you won't get much out of just a few hours there.
Do you have Rick Steves Italy? If not, get it ASAP. Read what he considers the highlights of each place, and decide for yourself if they sound like highlights to you. If not, substitute other things. Accept now that you will see a very limited slice of each place and will miss many things - and that that's OK.