We will be using the Venice/Corfu Ferry again in June. We use and recommend the Minoan ferry. This is a very nice and very large ferry.There is also an ANEK ferry, which I've never used. They all take approximately 24hrs to get to Corfu.
Confusingly the ferries often go past Corfu and stop at Igoumenitsa first and then return to Corfu. (its all Greek to me). Always check with the pursers office if Corfu is the first or second stop.
These are the only two ferry companies that go into Corfu harbor from Venice. From Ancona the ferry stops at Igoumenitsa where you have to change to a local ferry to Corfu.
Ferries operate from Bari and Brindisi to Corfu but then its a long drive down the Italian coast. I haven't used them so unable to compare.
From Venice you get fantastic views of the city from the top decks during the first 30 minutes of the voyage. Don't miss it! I would definitely recommend a cabin, though it does add a bit to the final cost. If you have a suitable vehicle its possible to camp on a car deck.
In my opinion you should pre book to ensure you get a cabin or car deck, but in anycase at peak season it's a must. You didn't say when your traveling? The Venice sailing times are not the most convenient, as most sailings depart at 1400hrs for which you MUST be there two hours before (1200hrs) if your in a vehicle. On arrival at the ferry terminal take your documents to the admin building and register for the trip. You get a card displaying your destination (Corfu) for the car.
All passengers are asked /told to walk on board only the driver can stay with the vehicle. So car passengers can get aboard before the car is loaded, though don't forget your passports and tickets!(ensure the driver has his ticket and passport). Agree to meet up with the driver at the pursers office where they give out room keys and information. Make sure you have all the items you want on the trip handy before arriving at the terminal.
There are several internet sites for booking ferries try www.viamare.com.
For your dog there is a doggy village and is at the top of the boat in kennels, for peace of mind take a padlock (bit bigger than a suitcase lock though). You can get access to your dog at any time during the voyage. Take appropriate items to clear up after your dog after walking him/her round the "usually" quiet open deck such as kitchen roll/polythene bags etc. There's a dustbin in the dog village for your use. I know some owners have smuggled dogs into their cabins!
Hope above of use and you enjoy the best way to arrive at Corfu. Let me know if you want further info?