Aside from Indonesia you might also consider Thailand, which has great infrastructure so it’s much easier to travel with young children. Phuket has by far the most options, and its rainy season usually ends in early November. For families I recommend Kata Beach and Karon Beach, which are both just south of the main shopping and nightlife town of Patong Beach.
Once again, it's worth noting that the first half of December will be cheaper and far less crowded than the time around Christmas, so if it's possible you should come early for best value. Still, compared to, say, Miami Beach, Cancun and the other Mexican resort towns are far cheaper and at least as fun as well. By the way, Cancun is mainly just a strip of high-rise hotels along a lovely beach, while Playa del Carmen (about an hour south) is a real tourist-friendly town with smaller hotels as well as hundreds of restaurants and shops to choose from.
Bali is wonderful, or at least parts of it are, but they now have a volcano issue and those flights are very long and never cheap. Machu Picchu is incredible, but it’s a place you visit for half a day or so, and they actually have their rainy season at the end of the year, so it’s actually not one of the better times to go. Actually, you might consider a trip to Argentina, which is warm that time of year, though rarely scorching. Buenos Aires is a wonderful city that is pretty good value now, and you can also visit the Mendoza wine region and Iguazu Falls and Bariloche, which are all even a bit cheaper than Buenos Aires.
The hotel rates all over Goa are highest in December and January, but in Anjuna or Vagator they still start at around £10 per night for a private room with ensuite. The cheap places are quite shabby, so it might be worth spending more. Vagator has an amazing beach, but it’s often filled with Russians these days. Anjuna is famous for its beach raves and music scene, but I’ve honestly heard nice things about many other beach areas in Goa.
Prices go up a bit when you go south into Malaysia, but it’s worth it if you’ve got the money. You might even consider Myanmar as well, which I’ve yet to visit, but I keep hearing great things about. It’s honestly the perfect time of year there. By April it’s insanely hot (the hottest month of the year), but December through February is lovely. It’s all quite safe as well, as long as you keep your wits about you. I’m happy to help with more information if you need it.
By the time November rolls around, Cancun has pretty much perfect weather for anyone. The heat of summer and early autumn has turned into a tropical warmth that is usually accompanied by pleasant breezes so it's lovely all day and all evening. The peak season doesn't start until late December so this is the perfect time to visit when rates are still nearly at their summer lows.
In addition to the obvious perks of pleasant weather and fewer tourists, off-season rates at accommodations are abundant. Voted No. 1 continental U.S. island by Travel + Leisure readers for the third time in 2018, Hilton Head has much to offer visitors. One of the most popular activities, perfect with autumn temperatures, is biking on the more than 60 miles of pathways covering the island, including along 12 miles of beaches. Bird watchers will keep busy spotting egrets, sandpipers, wood storks, sea gulls, pelicans, and ospreys. Well known as a golfer’s paradise, the Hilton Head Island area boasts more than 33 courses.
As for your Europe trip, I highly recommend spending 3 nights in any city you visit. If you change cities every day or every other day you’ll end up spending most of your holiday on trains or in airports rather than seeing the places you flew all that way to see. So if Amsterdam and Barcelona are musts, I would add exactly one other city if you’ve only got 9 days. Unfortunately it’s not on your list, but the best and easiest one to add between them is Paris. There is a train from Amsterdam to Paris in a bit over 3 hours, and another high-speed train to Barcelona from Paris in 6 hours 27 minutes with no changes. Paris is easily one of the most impressive European cities for tourists, but if you’ve been there before and don’t want to go again you can obviously choose something else.
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In November you can find excellent resort deals all over the Dominican Republic, but Punta Cana is typically the first choice, especially if you are looking for an all-inclusive resort with all the luxury extras at a great price. The resort is actually a collection of little beach areas on the eastern tip of the island, so it's very secluded and well away from some of the confusion closer to the capital city.
Trekking in Nepal during peak season might not sound like the most sensible choice for penny-pinching backpackers. But you can still enjoy the ideal weather without spending a fortune on in-demand hotels and expensive permits if you're happy to swap big-name trails, like Everest Base Camp and the Annapurna Circuit, for quieter treks to Kanchenjunga Base Camp and around the Langtang Valley. Idyllic conditions (dry but not witheringly hot) are also on offer in Rajasthan, one of India’s most vibrant and varied states. Chug between notable, attraction-rich cities like Jaipur, Jodhpur and Udaipur on an atmospheric and wallet-friendly Indian train, snacking on spicy pakoras and sipping sugary chai en route.