I don't move with my thread, but I'm getting a large collection. The drawers that Margo suggests are the ones that I use for my thread and it works well for me.
If you want to go more expensive, you could look at something like these thread satchels:
http://www.keepsakequilting.com/productdetail/9385/
CLEAR-SUPER-SATCHEL-trade;DOUBLE-DEEP-.htm
These might travel better because they latch, but you'd have to dink around with how best to store different sizes of spools.
Nancy