How would gravitational time dilation effects readings on clocks in deep space vs those on Earth...?

There is a time system called Barycentric Coordinate Time, TCB. It is the "proper time" experienced by a clock that is far from the solar system, but moving at a constant distance parallel to it. This clock is outside the gravity well of the Sun, Earth, and other planets.

TCB ticks faster than clocks on Earth by about 489 milliseconds per year. That means that if there was a radio signal from Earth that ticked at 1 pulse per millisecond, you in your spaceship would receive about 489 fewer tics per spaceship year. When you measure the length of an Earth year, you would get a value that is 489 milliseconds longer than the standard book value.

PS: since you are at a constant distance from the solar system, the light delay is constant. Or rather, the delay for the signal from Earth is the same at the beginning of the Earth year and at the end. So the compensation for this zero. If you really want to get picky, you could compensate for the delay caused when the signal passes close to the sun, Jupiter, or Saturn; this is done by scientists studying pulsar timing.

PS #2. There actually are radio signals that tic once per millisecond (Earth surface time). The most famous is WWV at Fort Collins Colorado, but I don't think the signal is strong enough to hear in your spaceship.

Space clocks tend to run a little faster, the difference in time between clocks in space and those on the ground are used to calculate your position on earth when you use a GPS device.

The extra aside the clocks are, the fewer the time on them will relate in any way. If we wager that "deep area" ability a great way far flung from noticeable gravity wells, we are able to anticipate a sign from that clock to seem to run extra right away than the interior of sight synchronized clock. we are able to anticipate this, by way of fact between the countless variables which could impression the plain fee of time, shifting deeper right into a gravity properly ability shifting extra slowly by using time. additionally, accelerating ability shifting extra slowly by using time, so that is attainable for the area-sure clock to seem to decelerate in the process the holiday, and then seem to hurry up as quickly because it arrives. This, all of direction, assuming which you're correcting for the sluggish-down that seems to ensue in the process the holiday purely do to pink-shift. i'm undecided what's meant to be lined once you assert "taking into attention the area-time continuum." in spite of each and everything, in case you're taking each and everything into consideration, the clocks are an identical -- in case you employ the clocks by way of fact the time-honored via which you degree all different issues. the element you're in all probability asking approximately is the calculation that GPS structures evaluate while monitoring the GPS satellites -- we ought to consistently alter for a fashion the satellites are shifting extra slowly by using time by way of fact they do no longer seem to be as deep interior the Earth's gravity properly as are the automobiles and drivers according to them. the effects of Einstein's theories had in no way been examined in exactly this variety earlier, so the designers of the GPS satellites and structures performed it secure. They made it attainable to envision what time it incredibly is in 2 strategies -- the Relativity-based answer, and the classical mechanics answer. seems, we'd desire to make the correction that Einstein's theory expected. Satellites for GPS could be "deep area" or they could no longer, counting on how deep your concept of deep area is going. yet, they are extra sufficient far flung from the middle of Earth's gravity properly than we are as to teach that the a number of rates of time are measurable, calculable, and critical. each and every working GPS proves that element ameliorations, in calculable strategies, in know to the two area and speed.