A chrysanthemum sold in flower is not automatically hardy in every garden. Keep its variety information and distinguish outdoor planting from overwintering a container under cover.
Plants that stay in the ground
In the mild southern and central regions, a border chrysanthemum needs no winter protection at all and can be left standing. Further north, cover the crowns, or lift the roots and heel them in a trench of soil through the cold months, then dig them out in early spring for potting up. One detail decides whether the lifted roots catch: chrysanthemum roots dislike being moved bare. Carry the soil with them, or the plant spends a month or more sitting still before it tries to grow.
Nothing needs feeding while it is buried. Wait for spring growth and start again from March.
Potted plants brought indoors
A pot of chrysanthemum that is to flower through the dull months wants fertile, loose material: garden soil and leaf mould with a little plant ash mixed through it. Feed up to the point where the buds are visible, then let the plant do its flowering on what is already in the pot.
Bring the pot in before the hard frost arrives rather than after it. Room temperature between 8 and 15 C is the working range, in the brightest place you have, with a small thin compound feed now and then. Cool and bright keeps the plant compact. Warm and dim gives soft stretch and usually no second flush.
Water and warmth in the cold months
Mulch and pot mixes that drain freely matter more over winter than any feed does; the mix described in repotting and pot soil is the right starting point. A wet crown through a frost is the common cause of a spring that never arrives.
A growing chrysanthemum in a pot is happy on a three to five day watering rhythm. Winter changes that arithmetic completely: evaporation is slow, the plant is barely drinking, and once a month is frequently enough. Count the plant's needs rather than the calendar.
Two positions to avoid, both of which look reasonable and are not. Do not put the pot in the blast from an air-conditioning outlet, and do not sit it against the radiator. Either one dries the air and pushes the temperature up in exactly the way that makes a cool-growing plant soft and unhappy. The winter discipline is the same one described for clivia in the cold months: cool, light, barely moist, no sudden heat.
What to expect in spring
Overwintered clumps push from the base, and the shoots coming up through old stem are the sign the roots took. Lift and divide at that stage rather than letting the clump crowd itself into a mat of thin growth. Chrysanthemums broadcast themselves in a bed that suits them, so a plant that has decided to seed about somewhere unexpected is a report on the drainage rather than an accident.
Feeding restarts with that new growth, not before it. Judging what the plant actually needs from the look of it is the quicker route than trusting a schedule. This article on sensible fertilising sets out how to read the symptoms.
Related reading: Can Chrysanthemums Flower All Year? Variety and Day Length.