Here is a 12 video playlist on relativity:
https://www.youtube.com/playlist?list=PLsPUh22kYmNAmjsHke4pd...
Here is a six video series on measuring space and time:
https://www.youtube.com/playlist?list=PLsPUh22kYmNBkabv9M4fX...
When you think about it, you will realise there’s almost nothing else it could reasonably be.
To figure out how long a time you traveled to get to my house, I need to know which path you took.
Our ordinary intuition is instead,
"To figure out how long a time you traveled to get to my house, I need to know the time on my clock when you left and the time on my clock when you arrived."
But that turns out to be wrong if you care about a high degree of precision or if you were traveling very fast.
Trying to stay alive in such a world would be quite a challenge.
You would see the image of a car before the actual car arrived
You seem to be confusing coordinate time with proper time. The time you experience as you get out of a gravity well is proper time, which, as a sibiling comment said, is something you obtain when you do the path integral of the path you travel. This time is not exactly affected by dilation. Anything traveling with you will not have its time go faster or slower. Anything that is not traveling with you, say stationary on Earth, will indeed have its passage of time shifted with respect to you.
Coordinate time is the time observed by an observer at the center of the gravity well, looking at you (not always, but generally so for most calculations. Also, observer = origin of frame of reference) and that is what is morphed and bent around. Since you're an observer as well, you also see other points experience time differently, since from your point of view, they're moving away from you.
That time is not constant on a scale that lets us measure it from space vs the earth much less from upstairs to downstairs is somehow deeply unsettling. If you're traveling at 10 m/s as you go out of the gravity well you go faster and faster because time speeds up, but your velocity per unit time stays the same.
Something that is really hard for my brain to wrap around in any meaningful way.
During some radio work this became actually a thing in that if you're doing time-of-flight measurements of originating RF energy and one receiver is in a platform flying above and one is on the ground, at high time resolutions you start having to correct for difference in the rate of time. This is definitely something you have to do if you want to get really accurate time from GPS satellites. So freaky it hurts.