For my main machine I divide my bobbins by thread weight. I use those terry cloth little girl hair tie bands around my bobbins to keep them from unspooling, and just toss them into little decorative bowls I keep near my machine - I have one bowl for my super thin piecing threads (Aurifil & Presencia) and then pretty much everything else goes into the other bowl. Those bobbins are all clear plastic so easy to see what color is what; I never seem to have trouble matching them up.
For my vintage machines with metal bobbins, I don't tend to keep as many of those wound up; those are usually just kept with the machine itself. I have five different types of metal bobbins between 17 different vintage machines, and I might use a spool of thread with any or all of those different bobbins, so storing the bobbins with the thread doesn't make sense for me.