You'd imagine an internet time (or civilisation time) used for timekeeping in interplanetary protocols, synced (relatively) well to Earth- or wherever the centre of human Civilisation ends up being. Because clock speed changes due to relativity, your Civilisation Time counter would have to take into account your position and velocity relative to earth to not slip- while syncing often enough to counteract This way you can have relatively stable time measurement, even when time strictly speaking passes faster for some computers than others.

Time slows down as you go faster. The people leaving earth are the ones accelerating, so they're the ones who travel faster into the future- less time passes for them. If we colonise stars around us, the people going to the further stars (assume all ships leave from earth) will lose more time: Civilisation Time will be ahead of the time they measure on their regular clocks.

Maybe we could use pulsars as clocks? I think they pulse pretty regularly.