Factory jack and lug wrench
Under the trunk floor or behind a side panel. Check it's actually there and that the jack still cranks before you need it at 6pm on a Sunday.
While the tire's still on the ground, so it can't spin.
Rated jack point. Spare on, lugs snug by hand.
Full weight down first. Star pattern, not circular.
I've changed enough tires by now, mine and other people's in school pickup lots, that the actual swap takes about fifteen minutes. The other twenty minutes people burn is finding out mid-shoulder that the wrench in their trunk doesn't fit their lug nuts, or that nobody's checked the spare's air in three years. So here's the whole thing in order: what's worth keeping in the car, the sequence that keeps you from stripping anything, where people mess it up, and what it actually costs to fix versus just replace.
Most cars come with the two tools that matter. There's really only one thing I'd add to the factory kit, and one small kit that turns a lot of flats into a five-minute fix instead of a full swap.
Under the trunk floor or behind a side panel. Check it's actually there and that the jack still cranks before you need it at 6pm on a Sunday.
The single upgrade I'd tell anyone to make. The factory wrench gets nuts on, it doesn't tell you when they're actually tight, and both over and under-torqued lugs cause real problems.
For a puncture in the tread, not the sidewall. Plugs it without pulling the wheel, then you air back up and you're done. This is the one that's saved me a full tire swap more than once.
Get these six in sequence and you'll never be the person jacking a car up on a slope with the parking brake off. Skip the order and you're the person with a tire that won't budge because it's spinning freely in the air.
None of these are exotic. They're the same handful of things over and over, and they're the difference between a clean fifteen-minute job and a tow truck call.
This is the part I actually track. A puncture in the tread isn't the same decision as a sidewall gash, and knowing the price gap before you're standing in a shop parking lot helps you not get talked into more than you need.
Temporary spare speed limits vary by manufacturer and are usually printed right on the spare itself.