When we moved to Guatemala, I had many of the same questions, so I thought I would share what worked for us. It will also help me remember what to bring in our upcoming international experience with 2 children this time.
1.When we moved to Guatemala, the kitchen things I missed the most were GOOD cooking utensils. Everything easily available in Guatemala was plastic that melted when it got hot. Yuck! I am glad I brought my good knives, and the mixer came in handy.
2. Since we moved when our daughter was barely talking, but planned to eventually homeschool- here's what worked for me with books: Between the ages of 2 and 3 there is literally a brain and language explosion! I was grateful that I brought books that could work for several age ranges including some that are good for beginning readers. Our daughter loved the Dr. Seuss, Curious George and things like that. I tried to mainly bring the ones that were big books with many stories inside to cut down on luggage. One thing is certain, no matter what you bring, it seems more books are always going to be in demand.
3. Something my daughter really likes are those magnetic drawing boards with the "pencil" attached and the little lever that "erases" the picture so that you can start over. I like it because it isn't messy and we can practice letters, drawing and storytelling. Besides this, we use books, a variety of snacks and that's about it. Guatemala was only a 3 hour flight for us, thankfully, so we haven't invested in electronic gadgets yet. Something I think she would really enjoy is an MP3 player with headphones since she loves music so much.
4. I wished I had access to more worship music in English in Guatemala, because I realized that my daughter was missing out on it and that she really enjoyed it when we came back to the States to visit. For her, specifically the simple worship songs work well since she's still learning the words. (Which I thought ironic since I hate worship music for the same repetition that makes it easy for her.) For you? I'm not good at suggesting music...
6. Right now we're really into tactile games like Jenga and blocks. But I like that we have Dominoes because it's really great for math/matching concepts(and making trains.) We also like Uno, Phase 10 and "Bancopoly;" the Guatemalan version of Monopoly. I look forward to when we can play games like Apples to Apples as a family. I have to admit, however, that no game beats just regular play outside with grass, dirt, pebbles and sticks and anything found in your new natural environment. Kids love nature (mostly) and can have a ball with the simplest items outside combined with a great imagination!
Hopefully, that was some help, though we have different kinds of kids and ages. I know that no matter how much you plan, when you get there you'll still realize something you'll have wished you'd brought. But it's OK. We're moms and improvising is one of our superpowers! ;)
Do you have suggestions for things that worked for you when moving abroad, particularly with children?