Basic entry tickets to the Vatican Museums have been selling out freakishly early this year. They aren't on sale yet for the period of your trip, but you should keep a sharp eye on the website (muzeivaticani.va) so you can make that purchase early, if you are interested in going there. The museums always seem to be a zoo, so springing for the higher-cost early admission tickets or an early-admission tour is recommended. Those special entries are likely to go on sale before the cheaper, standard-admission tickets. They're available (if not sold out) for late December, but they aren't available yet for January. I assume commercial tour companies (Walks of Italy, etc.) have their tours on sale already.
Also requiring ticket purchase before you arrive in the city are the Colosseum, the Domus Aurea and the Borghese Gallery--again, assuming you want to go to those places. The Colosseum tickets are tricky; scan back through the Italy forum here for recent threads on that topic. If you are unable to snag those online, there is the option of standing in line and buying in person, but I don't think we've had even one report here about how that goes from someone who has done it. The on-site sales started not that long ago.
For Venice, these are sights for which it would be prudent to buy tickets in advance: San Marco (long outdoor ticket line otherwise), Doge's Palace (especially for the Secret Itineraries Tour) and the Peggy Guggenheim Museum. The latter doesn't sell out early, I don't think, but you might otherwise encounter an outdoor ticket line, and who needs that on a short trip, especially in December?
Note that single-ride vaporetto tickets in Venice cost a very painful 9.50 euros each. I haven't been to Venice in the winter. I imagine it can be damp, chilly and breezy, so you might find yourselves happy to hop on a vaporetto rather than doing as much walking as you'd do at another time of the year. It's worth giving some thought to how much you're likely to use the vaporetti and considering purchase of a pass if that looks like a money-saving option. For example, if you want to go to Burano and Murano (and have time to do so), that will mean three tickets.
Prices for vaporetto passes are as follows:
24 hours: 25 euros
48 hours: 35 euros
72 hours: 45 euros
168 hours (7 days): 65 euros
I believe there's a better deal for children/students.