First of all, where are you? Are you in a climate where they overwinter, or somewhere you need to dig. Zone 6 and warmer they generally make it with good mulching in well-drained soil.

They multiply, but they' don't really "spread" since they don't have a mechanism to move by seeding, etc. as some bulbs do (at least never have for me). So you will need to dig and redistribute if you want more clumps.

You'll get lots of little "cormlets" from each old corm, but it'll be some years before they get to blooming size.

If you're below zone 6 and have to dig and store, you can save the bigger (or all of them if you're so inclined) of the cormlets along with the new corm - but, again, it'll be years before the little ones bloom for you.

I don't always grow glads, but when I do, I never bother to save them once I dig them up in the fall. They get composted.

If you do dig - cut the fans down to 5-6", dry them well on a piece of newspaper spread on the garage floor or just upright in a clean flower pot. If you store them damp, you'll get mold for sure. When dry, a light twist of the wrist will separate the old spent corm from the new one.

I've had relatively poor results getting a second year's bloom out of a stored corm despite having perfect storage conditions. Glads are quite inexpensive and readily available so when I want them I just start over. Gives me a chance to periodically change color schemes.

Others have fine success with storage - seems to be an "it depends" situation. Maybe some luck, too.