A core point of relativity is reference frames. From your perspective light in a vacuum always travels at c. The problem is that by special relativity, an observer moving at some speed relative to you will also see light traveling at c. This is a simple idea, but it causes a lot of very unintuitive effects. If we're being pedantic, relativity is very logical mathematically, but it's conceptually difficult because it flies in the face of how you think the world works.

Have a look at the simple inference example here: https://en.m.wikipedia.org/wiki/Time_dilation

Time doesn't necessarily slow down the further away you get from a clock. If you and a clock are both stationary (ie you're in the same inertial frame), you will observe it ticking in "normal" time, albeit delayed due to the distance. If the clock is moving relative to you however, you will measure its ticks to be slightly slower.

You may be confusing general relativistic effects which are distance dependent (as gravity weakens the further away you get).

If you carry a clock in your rocket, you will (in the rocket) measure it to tick once a second. When you get back to Earth, you'll find that it's lagged behind a clock that was started at the same time but was left on Earth.

Maybe have a look at simple wiki too https://simple.m.wikipedia.org/wiki/Special_relativity though it doesn't actually derive the Lorentz transforms unfortunately.

Ignore the gravity bit for now, that's general relativity and it's more complicated to explain.