Melbourne is a great city and having lived in Sydney for many years I definitely have the ability to recommend one over the other - but I won't. I will although point some of the lovely things you can do in and around Melbourne, especially with kids.
I have made this little trip across the bay before but the first time I did it, it was by accident. This time, we planned our little sojourn to Sorrento and back across the bay and it was delightful.
On this particular day one of our best friends was down from Sydney and we really wanted to show him what Melbourne had to offer. So we packed a few snacks and drinks to keep us going and drove down Melbourne's west to Queenscliff (which is a beautiful place in itself and deserves a more thorough investigation - mental note: add that to my must do list). The ferry to Sorrento leaves every hour from 7am to 6 pm (but may change during the year - so don't take my word for it, check before you head off) and it was a pleasant journey across the bay.
There is a place to buy coffee should you be so inclined and even a small kids play area on board to keep small people entertained. You can take your car with you (on the ferry) but on this day we chose to travel across on foot and just walk around Sorrento once we got there. The ocean views are beautiful and the ride was relaxing, even with kids (which is something you can't always say).
Seeing as we are still looking forward to a little bit more of this luscious hot weather - you may just have enough time to squeeze this trip in before it's layers, winter coats and I am recommending the best place to get a hot chocolate in Melbourne.