A furnace replacement runs in a fixed order. Some steps are trade work, some need your sign-off, and some need you to keep paper. Walk the job in this order and you will know what to look for at each stop.
Before the crew arrives
Confirm the start day and what the crew will need from you: clear access to the unit, the closet or room cleared, and the thermostat location known. Ask whether a permit is required for the work and who pulls it. Keep a copy of the quote and the equipment label details on hand.
Shutoffs and removal
The crew shuts off power and the gas supply before they disconnect anything. Watch that the gas shutoff is done before the gas line is opened. The old equipment comes out, and the old flue and any unused fittings are handled as the quote described.
Set, connect, and vent
The new furnace is set level and secured. The duct connections are made and checked for a tight fit. The flue is connected by the approved venting route, and the condensate drain is run if the model needs one. The gas connection and leak check are done by the trade, and you should see that check recorded as completed.
Electrical and controls
The power and thermostat wiring are connected. Confirm the thermostat works with the new unit before the crew leaves. Ask the crew to show you where the power switch and the filter slot are.
Startup check and sign-off
At startup, the crew should run the furnace through a full heating cycle and confirm that the burners light, the blower runs, and the flue draws properly. Ask them to check the airflow and to tell you what they measured. Before you sign, get the owner's manual, any warranty papers that came with the unit, and a written note of the startup check. Store them together with the quote.
For the typical service life of a gas furnace and when to plan the next replacement, see Gas Furnace: typical service life and when to plan.