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you either move through space or time at the extremes. in physical reality of course you move through spacetime.
if you move fast through space approaching the speed of light, you move slower through time, meaning objects around you experience you as gone for thousands of years while you will notice only a couple of hours.
and if you move not at all, your neighboring moving objects seem to speed up. for an outside observer you would seem to slow down, just like someone falling into a black hole looks to an observer as if it would take millennia to finish, but of course to the person falling in it's over much faster.
if you want a long live in comparison to the universe, you'd have to stand absolutely still. if you want to experience the entire universe coming and going within a human lifetime, move as close to the speed of light as you can.
but these effects only come into play at very fast speeds or very large mass differentials (since mass bends spacetime), you can not notice a difference if earth suddenly stopped dead in its tracks compared to its motion among the stars and the milky way, and assuming you magically ignore the breaking force.
