When one travels into space, time on earth moves faster than that of travelers, is this also the case when one travels from mars to earth?

Answer

The laws of physics always apply everywhere. The time dilation therefore also applies to a trip to Mars. But at the speeds we are talking about here, the effect is negligible.

As for Alpha Centauri. If we fly at a speed of 30 km/s, the speed of the Earth around the Sun, we will be back here in 80 thousand years. ‘We’ is often said, because at that speed the time dilation is negligible, so many generations pass. If we really want to ‘enjoy’ the effect of time dilation, we have to fly much faster, and we also return earlier. So we are talking about time scales smaller than 100 thousand years, and that is anyway short compared to the time that the Earth will last, at least as a planet.