I store quilting thread with all the quilting needles, etc. in a separate roll away in the quilting storage room. (This roll away has about 5 labeled drawers of various notions: painters tape, machine needles, pins, etc.) The embroidery thread is in its own small plastic tote along with the needles, etc.
The machine sewing thread is kept near my machine in the sewing room. My mother left me a lovely thread cabinet and it's not right at hand b/c the TV sets on it. The thread is kept clean and out of light. I have a basket on my sewing table with threads I'll be using for the project I'm working on. Bobbins are in the bobbin holders in that basket and other ones are kept with matching thread in snack baggies in the thread cabinet. This gets a jumbled mess of baggie plastic, but I can't think of another way of keeping those bobbins with the thread w/o spending $ on gizmos they sell. This all works for me. I'm not an avid quilter as some, but I need things organized!