Physicist: A little!
The movement of Earth, as well as the Earth’s gravity, change how much time we experience compared to other objects in the universe. If we were to occasionally compare our clocks to clocks in tight orbits around black holes or neutron stars we’d find they run slower than ours, and if we compare with clocks floating deep in the middle of nowhere we’d find that those clocks run a little faster than ours.
However, there’s no “true” time to experience; you can never experience time wrong. Time is relative which means that we can compare how time is passing for any two things, but there’s no ultimate “clock of the universe” to compare with. Your watch, no matter where you are or how you’re moving, will always read 1 second per second. That is, you’ll never see yourself in fast-forward or slow-motion. In that sense we can’t help but experience time correctly. Each of us may as well declare that our clock is the One True Clock, and everyone else’s is wrong.
Simply moving fast in a straight line isn’t enough to make your clock objectively run slowly compared to other clocks. If two folk run past each other they both see the other experiencing less time and, weirdly enough, this isn’t a paradox and they’re both “right”. There are two effects that do objectively (in a way that everyone in the universe can agree) cause clocks to run slow: the very poorly named “twin paradox” and gravity.
In spacetime the “length” (spacetime interval) of a trip is measured by a clock that makes that trip. It turns out (this is not obvious, but it can be understood) that the shortest trips are the ones that are the most circuitous. If you watch the ball drop on New Years and stay put for a year until the next ball drop, then you’ve made a pretty straight trip (in spacetime) between those two events. This path is straight, so it’s long, and your clock will read more. Instead, if you spend that year zipping around the solar system as fast as you can before coming back for the next New Years, then your path was decidedly not straight (in spacetime). This all-over-the-place path is short and your clock will read less. “The longest spacetime distance between two points is a straight line” may sound utterly insane, but it works. Long story short: if your trip involves a loop, then your clock is falling behind.
As it happens, the Earth spins on its axis and orbits the Sun and, along with the rest of the solar system and all the stars that we can see, orbits the galaxy as well. Each of these are loops, not straight lines, and each time the Earth makes one of these circuits it falls a little behind any clock that didn’t. This is a little hypothetical: in order to get a clock to sit in the same place while the Earth does an orbit to meet up with it every year, it would need a big rocket (it’s not orbiting the Sun, so it should be falling into it).
As fortune would have it, you can just use the gamma function, , to find the time dilation caused by running in a loop (for more complex paths, like those with different speeds, you still use the gamma function but you need calculus too). The velocity of the spinning Earth at the equator is about 0.5 km/s, we orbit the Sun at about 30 km/s, and the whole kit and kaboodle orbits the galaxy at about 200 km/s. The difference in time experienced between people living in Longyearbyen (near the pole) and people living in Ecuador (near the equator) is about one part in a trillion, which gives those proud Norwegians an extra second every 25 thousand years. Don’t spend that second all in one place, Norwegians.
The time dilation from the biggest of these speeds, our movement around the galaxy, amounts to one part in 4.5 million. That amounts to an extra second every couple months or an extra half solar year for every galactic year.
The second effect to consider is the curvature of space time caused by (or which is) gravity. Things that are lower experience less time than things that are higher. This can be explained (and even verified) by measuring how the frequency of light changes when it travels vertically in a gravity field. The details are terrible, but for most practical purposes (“most practical purposes” = “not black holes”) you can find the time dilation between two altitudes by figuring out how fast something would be moving if it fell from the higher to the lower and plugging that v into .
It’s reasonable to say that if you’re infinitely far away from something then you’re outside of its gravitational influence and your clock should be running “right”. If you fell from “infinitely far away” to the surface of something big, you’d be moving at the excellently named “escape velocity” of that big something. If you try leave a planet moving slower than the escape velocity, then eventually you’ll fall back. Excellent name.
To escape the Earth from the ground you need 11 km/s. More difficult is escaping the Sun (from Earth’s orbit) which requires 42 km/s. To leave our galaxy (from here) you need somewhere between 500 and 600 km/s. This time dilation from the Milky Way’s gravity has the biggest effect of those mentioned here.
The spinning of the Earth and the orbiting of the Sun do affect the amount of time you experience, but not by a lot. Despite being closer to the Earth than the bulk of the galaxy, it’s our orbits around, and position in, our galaxy that affects our experience of time the most.
By virtue of being a member of the Milky Way, we experience about 1 second per week less than someone hanging out deep in the intergalactic void. Most of that comes from the effects of our galaxy’s gravity directly; not from the motion of our planet.