Tools with gas engines need special care when stored for more than 2-3 months without use. Gasoline deteriorates with time. It turns into a gummy substance that will clog vital passage ways in the engine. Unthreaded gasoline will last about 6 months before the deterioration becomes noticeable. You can extend this time by adding a fuel stabilizer. These products are not snake oil. They actually work. Fuel treated with a stabilizer will last about 1-1.5 years.
Your gas engines need to be winterized to minimize damage due to condensation and fuel turning bad. Here are the minimum steps you need to take to winterize your engine. Other guides recommend things like tightening all fasteners, installing new air filters, or changing the oil. These steps are not necessary for winterization. They are part of general maintenance and do not have to be done. The steps below are the ones that are required.
First and foremost do what your manual requires. The manufacturer knows best. That being said, there should not be any special steps in the manual beyond the ones detailed here.
1) Manage the gasoline
You have a couple of options here. I recommend treating the fuel with stabilizer, topping off the tank, and running the motor for a few minutes. This ensures that the fuel stabilizer is pumped to all parts of the engine. The full fuel tank has no room left for air and water vapor. This prevents internal rust. Store the mower in this state.
The second option is draining all fuel from the engine. This is best done by running the mower until it runs out of fuel. Store the mower without any fuel in the tank. Keep the tank lid closed to prevent condensation from forming. It is unclear if this option is better than using fuel stabilizer. Both work well enough and there is no data showing one or the other to be superior.
Some people say that gasoline will keep over the winter; no need to pay for a fuel stabilizer. This is very possible since gasoline should keep for about 6 month. You can take this option, as long as you know that you are taking a small, calculated risk of the fuel turning bad faster. I recommend running the mower every month or so for a few minutes to at least get the fuel moving through the system.
2) Manage the oil
Simply leave the old oil in the engine over the winter. Adding new oil in the fall does no harm or good. It simply takes care of a maintenance step you should do once a year. It does not have to be done to winterize an engine.
Some people advocate removing the oil and storing the engine without oil. I recommend not doing this. The oil coat on the internal surfaces of the engine prevents rust from forming. In reality most surfaces in your engine will still be coated with oil even after draining the reservoir. This option is probably just as good as the leaving the oil in the engine.
Just leave the oil alone and change it in the spring to start the new season.
3) Manage the battery
Most push mowers don't have one, so there is nothing to do here. If your mower has one, it is almost certainly going to lose its charge over the winter. Take it off the mower and store it in a warm well ventilated place. Hook it up to a device called a battery tender (sometimes called a trickle charger). This device will maintain the charge in your battery.
New batteries often make it through the winter without losing charge. Don't expect this to be the case when the battery gets older. Taking the battery off the mower will extend the life of your battery significantly.
When batteries fail they can emit toxic gases. It's not wise to store your battery in the living area of your house. Pick an out of the way place.
You are done. Take care of the gas, and maybe the battery. Everything else is extra work and does not have to be done today.