Venice is a wonderful italian choice, but to really boost it the right way the right beginning strategy is the key. So, to grow fast and to easily conquer territories you have to pay attention to a couple of not so visible elements:

1) you are compelled to control narrow long territories, such as the eastern Adriatic coast, and narrow territories mean both difficulty in mantaining a quick reinforcement chain and frequent attacks on long exposed borders. Magyars, HRE and Byz could become annoying reasons to waste unwanted money to face their attacks time after time.

2) in the beginning Milan can be a quite ally, but if you really choose to ally with them and to forget them for many turns, you are in the meantime letting them build a strong economy and very dangerous armies. You risk to pay a lot, later, for such a lazy choice.

So, that's what I did:

Destroyed buildings in Durazzo and disbanded troops down there, to get an initial boost of money. Didn't care of losing the town. It will be so easy to take it again when needed lately in the game...

Build up a little but hard army in Venice, sent it with 3 spies to Genoa to open the gates, conquered the town and made a ransack. Got lot of money and strong happyness. Raised taxes. Build a harbour. Made a new little strong army in Venice. Same tactic used to Bologna, that already has a harbour. Next turn got Florence from rebels the same way. In the meantime Pope asked me just to take Florence, that I was already taking, and got some free troops as prize. Funny. Same tactic again to get Milan. The Milan remaining armies (only 3 and not so strong) became rebels the next turn, and they sadly faced my towns being beated with easy.

End of Milan faction. Northern Italy is mine.

Now spent some turns to build up my economy and enforce my armies and garrisons.

After a short body-building time, sent spies to Naples, followed by a strong army, all along the Adriatic western coast. Faced Sicilian king's army in the south and beated it 2 times, then finally arrived to the gates of Naples (opened the usual way) and conquered it. Usual ransack and usual lot of money and happyness.

Pope asked to take Rhodes from rebels, so I build up an army in Crete, sailed to Rhodes, put it on siege, and taken it after 1 turn. Another price from Pope. Great!

Then used few more turns to enforce economy all over my territories and to build up a strong army in Naples. When ready, sent usual spies to Palermo and taken it with the new army. Ransack, money and happiness the usual way. Sicilian faction is out. I am the owner of all Italy, and only have a precious tenant in Rome (the Pope!).

Now, it's time for greek territories (Corinthus, Thessalonica and Byzantium, plus my former town Durazzo).
I did it my way

Next move: Asia.

But the key of success is to be found in the beginning attack strategy.
In my way, I have the strongest economy of the campaign, and the best manageable territories, in a very smart blitzkrieg.

See you soon, mates!