I don’t drive, as it’s been largely both unnecessary and impractical with where I’ve lived in the past decade, but I know how to check and change the oil in a car, among other basic vehicle maintenance tasks. It’s not hard or time consuming.
When I owned cars I often used to fix issues myself (as I used to drive old ones I've replaced entire window mechanisms and wiring harnesses) but not the oil changes. It's just not worth it. And it's so much easier with a full hoist, I only had portable jack.
As for disposal: in a fair few places I’ve lived the city has had regular free pickups or drop off locations for “hazardous” domestic wastes such as oil, car batteries, etc.