The box says it is a simple job. It is, when the wires match the instructions. When they do not, it becomes a long afternoon. Here is the order to work in, so the furnace and your mother's dinner both survive.
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.
Before you touch anything
Turn the system off at the old thermostat, then cut power at the breaker or the switch near the furnace. If you cannot find that switch, stop and call someone who does this work for a living. Take photos of the old wall plate from three angles and write down which wire goes where before you pull a single one loose. The old plate is your map. Keep the photos on your phone and in the shared folder so the next person can find them.
Check the wires and the common wire
Most thermostats use a set of color-coded wires that match letters on the terminals. Check whether the new model needs a common wire, often labeled C, and whether your mother's wiring already has one. Some homes do not. Read the spec sheet before you buy, not after you have driven the box back to the store. If the wires you have do not match what the new unit asks for, that is a decision to make before the install, not in the middle of it.
Mount the base and connect one wire at a time
Level the base plate, screw it into the wall, and connect each wire to its matching terminal, checking your photos after each one. Tug each wire gently to make sure it is seated. Do not leave a loose wire tucked behind the plate. The next person to open it should see a clean connection. Put the cover on only after you have checked every connection twice.
Power on and pair it
Turn the power back on at the breaker. Let the screen wake up, join your mother's Wi-Fi, and follow the app's prompts. This is the step where most people stall. Decide before you start who owns the account. A workable choice is a family email address that more than one person can reach, not a personal inbox only you can open. Save the login in the shared place you both use, and note the device name in the app so you can find it later.
Test every mode before you leave
Run heat, run cooling, and run the fan on its own. Check that the system switches on and off at the settings you chose, and that the display matches the temperature in the room. Leave the house at a comfortable setting, not the extreme one you used for testing. Then sit with your mother for a few minutes and show her the one button she needs. A simple setting she can read is worth more than a clever schedule she cannot find.
Decide who gets the alerts
Many apps send notifications to whoever is signed in. Decide whether those go to you, to your mother, or to both, and whether anyone will read the alerts during the afternoon. A notification nobody reads is just noise on the phone.