I've been to a lot of the stans although 10+ years ago. Uzbekistan would be the easiest to visit, but it depends whether that sort of historical type thing appeals to your kids.
I haven't been to Kazahstan which I think is also relatively easy, but perhaps less to do and more expensive.
Kyrgyzstan is a bit harder to visit as it is mountainous and quite rural making getting around more challenging. It would really depend on how you plan to go - if on an organised tour perhaps you won't have any logistical issues and you can simply enjoy the mountains. Every village has some dedicated tourist lodging that meets a certain (basic) standard and has an English menu, so you can stay with locals and really experience the country. On the flip side, I remember arriving in a town at about 6pm and trying to find somewhere to eat. The market had closed, so nothing was available. I had almost given up when some random man came up to me and said in broken English "I hear you are hungry. Come. My wife will cook". It was a great experience and shows the hospitality of the region (which is incredible), but it was also hard - three hours of sitting and chatting before the food arriving. Overall, I would say it was doable for a family if you take some taxis and have a relaxed itinerary. You initially mentioned trekking though and I would be inclined to go to Nepal as a starter for that. Maybe an organised trek would be doable, but there are very few facilities in the mountains. I just did a couple of nights in a yurt rather than a trek.
A more wild option would be to travel from Dushanbe (Tajikistan) to Bishkek along the Pamir highway. This is a pretty special trip if you like mountains, but Tajikistan is even more rural and difficult to get around than Kyrgyzstan, so you would need some patient kids. The Pamir Highway was probably one of the most memorable trips I have done, but I'm not sure I'd recommend it unless you have a lot of travel experience.
Other places that you might want to consider: Jordan for a quick trip, Zambia/Malawi if you want to use public transport, Peru for some Incas and jungle, Brazil (although quite big if you only have two weeks), Mexico (pretty straight forward), Cost Rica, Malaysia (easy to get around), Philippines (this has far fewer tourists than the rest of SE Asia - would love to take the kids to Sagada along with a number of other places), Kerala in India and Sir Lanka.
I'd like to try Bhutan.
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