Daylilies are like any perennial namely that they can easily be divided, either in spring or fall. Just dig the whole clump, take a sharp knife or spade and cut it into several pieces (you can get quite rough witht them, they'll be fine). I find there's no need to wash off the dirt and just replant seperately to your wishes or discard/give away the excess.
Also any faded flowers are best picked off unless you want to sow your own seed. Then you just leave it and let the fruit (seed pod) develop until the seeds are ready. Mind though that they rarely come true (meaning the offspring is exactly the same type/cultivar as the parent plant) and you'll end up with a mix.
Division as explained above WILL produce true plants since they're clones from the original.
