In SE Asia you have many choices, but probably the best choice for a family would be Phuket, Thailand. It’s more expensive than Goa, but the infrastructure there is much better. Patong Beach is the biggest town on the island and the main tourist hub, but for a family I’d recommend Karon Beach or Kata Beach, both of which are just a bit south of Patong, and both lovely.
Pretty much everywhere is going to be at or near capacity for those dates because half the world is on Christmas break. Between Australia and New Zealand, I think New Zealand would be far more interesting for 10 to 14 days. In Australia you pretty much have to spend a few days in Sydney, a few days in Melbourne, and a few days at the Great Barrier Reef, all of which require flights and will be very crowded over those weeks.

Yes, these lists are reviewed and updated at least once a year. I’ve been adding new cities each year and even removing a few that have become too expensive to be on a “cheap” list. Personally, I’m an obsessive travel planner (which is why I do this for a living) and I like planning at least a few months in advance. You usually get the cheapest flights around 3 months out, although it can range from 2 to 5 months for longer flights. In other words, now is a great time to plan a November trip.
Rio is probably the most beautiful city in the world as far as the setting is concerned, but it’s also famous for petty crime. As you mentioned, I think it’s quite easy to stay safe as long as you are aware of the situation. The biggest problem by far is tourists getting robbed by someone holding a knife or sometimes a gun, and they are often in groups. However, this almost always happens to people who are walking close to the water on the beaches after the crowds have gone for the day. The sidewalks along Copacabana and Ipanema are brightly lit and safe, but there is a mound of sand between the sidewalk and the water, so if you go to the water you can’t be seen by the security guards and police on the street. As long as you avoid that, and any other situation where you are alone in a dark place late at night, you’ll be fine.

Sprawled across two countries (Chile and Argentina), Patagonia is one of nature’s last frontiers. For trekking enthusiasts, adventure seekers and nature lovers alike, this remarkable landscape of snow-capped mountain peaks, imposing glaciers, gushing waterfalls, and beautiful national parks is the very definition of ‘epic’.  It’s also one of the most isolated territories in the world, with roughly two people per square kilometre. Here at the end of the world, our planet is at its most wild and spectacular.
The subtropical climate of Hong Kong means that summers are particularly hot and wet, and spring is warm and humid. November is the sweet spot, with clear, sunny days and mild temperatures. During your trip, you'll want to take a traditional junk boat ride on the harbour (seen here) and check out the sweeping city views from Victoria Peak, which you can reach by tram. Another can't miss? Stockton, a hidden speakeasy-style whiskey bar in Central.
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