You need to find out if you have "hardy garden mums" or "florist mums" then find out if it does produce viable seed, some hybrids don't, otherwise you might be able to take cuttings or increase it from division in the spring.
The center of the flower is where the seed head will form; this will dry and turn brown. Once dry cut the seed head from the plant, place it in a paper bag with the top open to continue drying before separating seed from chaff. When that is done, place seeds in a clean, dry jar with a desiccant pack (like the ones in shoe boxes or medicine bottles). Some gardeners just crumble the whole seed head over garden soil in the spring.