I took 2 young children backpacking round SE Asia about 8/9 years ago! Just me to start with, then met up with a friend after 2 weeks, and had another week in all together. I had an AMAZING time.
Kids were 3 and 7 at the time. We did Singapore, Kuala Lumpur and Borneo. I picked 4 destinations, and we stayed for 3-5 days in each place. So we ended up with 2 cities, one beach and one jungle, and then the odd day in hotels travelling between.
In Singapore we stayed in a Youth Hostel, in KL we stayed in a super fancy Air BnB with an amazing rooftop pool. Then we did a jungle camp in Borneo (slept on wooden platforms in the open air!), and a nice beach side home stay type thing, also in Borneo.
It was pretty intense to be fair, but was so much fun.
Best things:
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The people. My 3 year old in particular was adored by people every where. Like, genuinely adored. It felt so so safe.
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Kids highlight was Legoland Malaysia 😂
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My highlight was the jungle camp. We went here: https://www.uncletan.com/ We saw unbelievable things - crocodiles, huge bats, monkeys jumping into water and swimming, orangutan in the wild - a male and female actually doing a sort of mating ritual (not the actual mating....) It felt like a real adventure.
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KL fancy apartment
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Snorkelling in the South China sea with my 7 year old, swimming over fish and urchins - it is still one of my favourite memories of us together, and he is now almost 16!
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Our first stop was Singapore, which was such a perfect introduction for me. I had never been outside Europe before, so it was all very new! There was plenty to do for all of us, including well into the night, which was helpful for jet lag. And lots of food options. I still remember the first meal we all got, both kids chose this Asian soup with dumplings, and I had peanut butter on toast! They led the way for me really.
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It was really quite cheap, once we got there. And the flights were not too expensive. But... 9 years ago.
Hardest things:
1)The heat. 3 year old struggled. A lot.
- Intense being a just the 3 of us 24/7 with no break. It was great when we met my friend, because I could leave 3 year old with them, and take 7 year old out alone (like snorkelling!). Before that, we could only do what 3 year old could do (obviously). I have just reread and seen your youngest is 9, so I guess the same theory applies, but there is a lot more a 9 year old can do, so this might not be a problem for you.
I can't think of any other negative! Maybe other people's reactions?? I got a lot of "what's the point? They won't remember. Why bother? It'll be hard work" etc. Well, they do remember! Even the 3 year old has some memories of the trip. And even if they didn't - so what? I went because I wanted to go, I was newly divorced, and I happened to have a few weeks window with no work, and it felt very much 'now or never', and I had an amazing time! And so did the kids. And yeah, parts of it were hard work. But so would being together 24/7 at home. Again, with the age of your kids, you might not get that negativity.
Their dad was very supportive of it incidentally, and encouraged me to go. We just did a few video calls, and all was fine.
We went over Christmas time, so school/nursery wasn't much of an issue. Just missed a few days either side.
Luggage - I had a large backpack, 7 year old had a backpack, and 3 year old had a tiny backpack. I also took a side carry sling thing which was very useful at times, when 3 year old fell asleep. I also had a big karabiner clipped to my backpack, which I attached her little backpack to when she couldn't manage it.
I know my 2 weeks alone isn't quite the same as your 3 months alone plan! But just writing to encourage you to go for it really! And I seriously recommend Borneo.
Have an amazing trip!