Chrysanthemums are perennial Ourbilly. You don't say whether your plant is being kept indoors or out.

If it's outdoors in the garden and used to that situation, just leave it alone and cut down the finished stems in spring when it starts to shoot again from the roots.

Chrysanths that are treated as pot plants can be grown in the garden afterwards, but if it's indoors in a heated room now it would get quite a shock if you were to put it out into the cold, cold snow when it finishes flowering.

If you'd been given the plant in the summer I'd advise you to plant it in the garden when it finished flowering, but if you do that at this time of year the contrast in temperatures may well kill the plant. I suspect that there isn't really a good option which is why no-one answered your first question.

My best guess is keep it indoors all winter, even after the flowers finish and plant it out in spring.

They are perennial & will live for years in the garden flowering every year.

My wife was given one a week ago. It's now in the kitchen. When the flowers finish I'll cut them off & then put the plant in the window till spring. Then I'll put it in a bigger pot & put it outside on my balcony.

If it's a pot plant then you will have to keep it in a cool, frost free place till spring is well advanced before planting it in the garden. Even then you must accustom it to the lower temps outside - called "Hardening off" - before you can plant it. That means putting it out in sheltered spot during the day & bringing it in at night. After about 2 weeks or so you should be able to plant it in the garden.