But nobody in my family had any gas-powered motors for anything at all.
I'm 61 now, and a volunteer firefighter. We have lots of gas powered chainsaws, circular saws, ventilation fans, and more.
I'm still extremely uncomfortable with starting these engines.
You cannot live this way. I can walk you through anything related to home care.
I don't have any data but I'd assume it's vastly safer to mow the lawn than drive a car.
I would guess lawn mowers are higher risk per time or distance, but lower prevalence in injuries per year. I would guess the injuries are mostly a) hit rocks, b) did something dumb with the mower, or c) general outdoors risks like slipping or being bit by snake, etc.
Put gas in it. If there's a soft rubber thing near the gas, hit it twice to provide some fuel but no more as you risk "flooding" the engine.
Hold down any handle at the top of the mower, often the thing will require you to manually hold it down during start and all operations.
Look for the starter pull. It's often on the right, on the motor or mower handles. It's a piece of plastic attached to a cable. Give it a yank with a full follow through. It doesn't have to be maximum effort but too gentle won't work either.
Though personally I'm a fan of "kill your lawn" efforts. You can smother it with cardboard (or burn it, or till it, etc) and replace with native meadow.
(Has cleared overgrowth consisting of both vines and grasses roughly a meter high give or take some)
What you do is tip the mower up, holding the handle near the ground and push it right into the mess. Then lower it down, essentially taking a "bite", which will cut many folded over plants.
Pull back, then tip and advance repeatedly, cutting more each time.
I cleared a quarter acre this way. Took one hard afternoon and a couple tanks of gas.