You will probably find the filter at the worst possible moment, which is when the house starts sounding tired and you are standing in the hallway at Mom's with the car running in the driveway. So let's do it on a calm afternoon instead, while the kids are at the bus stop and nobody needs a snack.
Local option: New Plan Care gives families in the St. Louis area in-home help so a parent can stay home: companionship, help with bathing and dressing, meals, and overnight care. Call (314) 405-0887.
Find the filter before you shop for one
Most filters sit in one of two places: a slot in the return air duct, usually near the furnace or air handler, or a narrow slide-out panel on the furnace cabinet itself. Some houses have a grille in a hallway or a ceiling corner that hides the filter behind it. If you can't find it in five minutes, ask the person who last serviced the system where it lives, and write that answer on the inside of the closet door or on the card you keep with the paperwork.
Turn the system off at the thermostat before you open anything. It takes a few seconds and it keeps the fan from pulling on your fingers while you work.
Read the size printed on the old one
The size is printed on the cardboard edge of the frame, usually as three numbers: length, width, and depth. Write them down on your phone right then. Don't trust your memory, and don't trust the size you think the house uses, because older systems sometimes take a size that isn't standard. If the print is worn off, pull the old filter out, take it with you, and match it in the store by holding it up to the shelf.
The depth matters as much as the length and width. A filter that is a half inch thicker than the slot will not fit, and a thinner one will let dust slip around the edges. Both problems look fine from across the room and both cost you on the next cleaning.
Check the arrow before you pull the old one out
Look at the frame. You will see an arrow printed on the side, and it shows which way the air should move through the filter. Take a photo of it with your phone. The arrow should point toward the furnace or the blower, which is away from the return grille and toward the ductwork. Once you pull the old one, it is easy to forget which way it faced, and a filter put in backward still looks installed, which is the trap.
Swap it and check the air
Slide the old filter out slowly so the dust stays where it is. Note what you see. If the filter is dark all the way across, the house has been working hard, and you may want to check the schedule more often. If the filter has a clean band and a dirty band, the air may not be sealing well at the slot, and that is worth a call to whoever services the system.
Slide the new one in with the arrow pointing toward the blower. Close the panel or door until it clicks or sits flush. Turn the system back on at the thermostat, then walk to the nearest vent and hold a tissue near it. If the tissue moves, air is coming through. If nothing moves at all, something is off with the panel, and it is time to stop and call instead of forcing it.
Decide who keeps the spare
This is the part that makes or breaks the job. A filter you own and a filter that lives in the closet at Mom's are two different things. Buy a spare and put it where the next person will find it, not in your trunk. Then answer three questions on paper or in the shared notes, so nobody has to guess later:
Who checks the filter, and how often? Use the schedule on the package or in the system's manual, not a number a friend read somewhere.
Who buys the next one, and where do they buy it? Write the size on the note so the purchase is easy for anyone in either house.
Who calls the service person if the system sounds wrong, smells wrong, or stops? That phone number belongs on the same note as the size.
If you do this on Mom's row of the split week, the filter becomes one of those small tasks that nobody has to remember, because it has a place on the calendar and a name beside it. That is the whole trick. The filter itself is easy. Keeping track of who is responsible for it is where most families get stuck, and that is exactly the part worth writing down.
If the system needs more than a filter change, the repair belongs with the professionals who service it. You can find the local options listed on Local Help in St. Louis, and you can start with whoever installed the system in the first place.