Italy should be at least a bit warmer than England this time of year, but not by all that much in a typical year. Also, if you only have about 9 days I think I’d try for no more than 3 or perhaps 4 total cities (Venice can be appreciated in about 24 hours if you are in a hurry). And of course, Paris and Amsterdam will both be quite chilly, so I think I’d save those for another trip. My recommendation would be to spend the whole trip in Italy so you aren’t so rushed. Do Rome for at least 3 nights and Florence for 2 or 3 nights, and then Venice for 1 or 2 nights. You could still have time to pop down to Sorrento for a couple nights to see Naples and Pompeii and whatnot.
Of course in November you can pretty much forget about Europe if you want warm weather and little chance of rain. Fortunately you can go almost anywhere in Latin America and through many parts of Southeast Asia on a cheap flight, and you'll be in the sun with shockingly low prices, easily making the flight duration and cost worthwhile. The best holiday destinations in November are found all over the world, which you'll see on the list below.
Yes, I’ve been to all of the Nordic countries, but mostly during the summer months. However, I know that those Northern Lights tours are popular and well established. On one hand, you know you’ll be visiting areas where there is only a bit of sunlight in the middle of each day, so it won’t be great for general sightseeing. But the cold months are also the best Northern Lights months, and if you want to see them then it’s your best bet.
Blisteringly hot in summer, Tokyo is a breeze in the cooler months. Fight the dizzying jet lag and head out for the world's best sushi or ramen, shop in the city's amazing department stores, and visit the parks surrounding the Imperial Palace in the center of the city. Escape the crowds somewhat by heading to Yanaka Ginza, which retains a bit of old Tokyo and has plenty of places to eat and shop. Do make time for a side trip to Hakone, where you can visit a Shinto shrine and see magnificent Mount Fuji on a clear day.
Fortunately, it rarely rains at all in Cairo, so even though December is one of the wetter months, that means nothing and you are unlikely to see even a drop of rain. Hotel prices do reach their peak in Cairo starting late in December, so it's best to go early in the month if possible. Egypt is definitely one of the cheapest places to travel in December, and the weather is surprisingly nice as well.
There’s really nothing like the holidays in Manhattan. December tends not to be too cold in New York -- you may actually pine a bit for snow as you mosey past Fifth Avenue shops, peering in at the elaborate holiday window displays. The rink at Rockefeller Center gets crowded, but catch it midweek and you’ll have room to skate a lap beneath the most famous Christmas tree in the world. And if you’re still picking up last-minute gifts, you really can’t beat the shopping in this town. Even usually gruff New Yorkers seem to have a brighter spirit this time of year, and carols echo through the subway stations thanks to the city’s many transit musicians. There’s no better time to go and bask in the Home Alone 2 Christmas vibes.
Since you might be open to flying then another nearby option to consider is Siem Reap in Cambodia. It’s a pleasant tourist town on its own, but it only exists as the gateway to the amazing Angkor Wat temples nearby. Those temples are by far the most dramatic and memorable tourist attraction in southeast Asia, and I think your daughters are old enough to appreciate it all. But again, the town of Siem Reap is nice with plenty of other things to do, so it’s a great all around stop.
Similar to so many on this list, Cape Town begins its busy season in the second half of December, so November is ideal in that the weather is nearly perfect and you don't have to worry about high-season crowds or prices. This is a city that has things available at almost any price level. Many of the hotels cater to the 4-star resort crowd, but there are also plenty of hostels and cheap hotels so backpacker-types should be able to get by on a low budget.
Costa Rica is kind of the Thailand of Latin America in that they have great tourist infrastructure and it’s the easiest place to get around. They also have a lot to offer with the beaches on both coasts and many natural sights and activities, plus of course the Arenal volcano, which is a really nice area. San Jose is best as a place to just pass through on your way somewhere else.
Traveling around India is a shock to the system, but it’s not as bad as it looks. If you pay a bit more for the AC3 class on trains (air-conditioned, 3 bunks per side of the compartment), or AC2, it’s pretty comfortable and civilized. Mumbai is worth a couple days before you’ll be desperate to get out, but many parts of Rajasthan are much nicer and fascinating.
My other two suggestions are San Juan, Puerto Rico and Cartagena, Colombia. Both cities have really wonderful colonial districts and modern resort districts nearby. San Juan is more expensive, but still reasonable, and it has far better beaches. Both have a bit of rain in November, but it tends to come and go quickly. You’ll get similar rainstorms in most of the Caribbean in November, so it’s a bit tricky. I hope this helps. If you had something else in mind let me know and I’ll try again. -Roger