I keep thread in plastic storage boxes. I bought them at Walmart in the toy department. They are frosted clear plastic and are meant to hold toy cars like Hot Wheels. The size is perfect for threads, even embroidery threads. They stack together nicely and are only about $6 each. A lot cheaper than similar boxes sold for thread specifically.
For bobbins, I use the Bobbin Saver. I use a different color for the different machines so the bobbins don't get mixed up.