If you are flying you are going to have limited space. My kids have one of their lamps that is called Lappan, which is great for a light and only costs $5. They fight over where it should be located; I wish I got 2.
I also wished I picked up one of their adjustable led lamps, either clamp or lampstyle, to place above their wardrobe for lighting when they are picking out clothes. I would also buy the stick on mirrors so they can have a mirror inside their wadrobes for checking themselves.
I will have to disagree about Ikea. With only a couple exceptions, we went through a long period of time using only hand me downs, thrift, or doing without. It was a sheer pleasure and luxury the first time we went to ikea and pick out furniture of our own, that was new, and actually fit the space/function it was meant to for the room. We have a poang chair in our bedroom and it is very comfortable. The lack tables are very useful too (but too big for a suitcase).
Don't just buy things to buy things. Buy things because you have been looking for months and cannot find a very specific thing on the secondary market, and you know the dimensions etc it needs to be. As for frames I've never had problems getting all the frames I needed from thrift stores (may need to throw out some artwork to use).