Start with the power. Turn off the furnace or air handler switch and the breaker that feeds it. Then take the old thermostat cover off and look before you touch anything.
Photograph the wall and the wires before you disconnect a single one. Note which terminal each wire sits on, and write down the letter printed beside it. A clear photo is the record you will use if the new unit's terminals do not match the old ones.
Identify the system type. A heat-only system, a cooling-only system, a heat pump, and a dual-fuel setup each want different terminals. A heat pump thermostat usually needs a terminal for the reversing valve and may need one for auxiliary heat. Check the new unit's manual for the system type setting before you wire it. The Heat Pump page on this site describes the equipment you are likely to be controlling.
Check for a common wire. Many newer thermostats need one to stay powered. If the wall has no common wire, the new unit may need a different power route or an adapter. Do not guess at that step.
Read the warranty paper for the equipment before you begin. Some terms ask for a trade installation or a written record of the work. Others say nothing about it. Read the paper rather than assuming either way, and keep the photos with it.
Call in a trade if the wires are scorched or brittle, a terminal has no letter and no clear match, the old unit used a line-voltage connection, or the new unit needs a common wire the wall does not have. Those are the points where the job stops being a swap.
When the wiring is done, restore power and test each mode in turn: heat, cool, and fan. Confirm the system responds to each call. If any mode does not respond, shut off power again and stop. Record the result and the date in the folder.