I'm afraid this one is kind of a yes or no question, because I think I would get lost in the discussion of metrics, etal. As a layman, I get that if you travel fast enough (by accelerating and decelerating of course), that when you get to a destination star, less time has passed for you than a person that stays behind. (ie. you could send a light speed message back to Earth 10 light years away and it would be received by someone say 1 year older than you would think they would be (adding the message travel time) from looking at your clock when you send the message). Hopefully, so far so good. Now you leave someone behind at the first star with a clock and do the same thing with the same result. Then, you do this again and travel back to Earth. Shouldn't this time dilation effect be additive or does it get crazy because I am adding a second space dimension to the metric?
Ok, so the main paradigm you need to apply is inertial reference frames.
Each "location" in your system is itself a reference frame, may it be earth, some distant planet, or your imaginary spaceship.
Now each of these reference frames is contained within a larger reference frame, for the planets it is initially their solar-system and subsequently their galaxy and finally the universe. This total may be considered stationary...*
So on earth, we have a given constant velocity relative to the universe. The subtle variations in this velocity due to rotation of the earth and the velocity of our planet through it's orbit are considerable, and at the moment negligible.
(Here we can consider the planer nature of orbits in the context of the temporal eccentricity introduced by orbiting with general velocity's projection maximized against the orbital plane.)
Meaning that more or less:
On earth clocks thick at a "constant" rate. lets call this 1hz. By doing this here we create a metric, by which we can measure the passage of time in alternative reference frames.
So on another planet P our inertial reference frame has a different velocity, relative to the universal reference frame resulting a second rate we can call p hz representing the rate of time experienced at P.
Now for or space ship we will assume for simplicity that we start in one reference frame, blink to a velocity towards another reference frame and stop relative to that reference frame upon arrival. and while you travel the ship is a reference frame with velocity v and therefor a temporal rate s hz < p , 1.
Now we will introduce a variable t for the time passed on earth as we conduct these though experiments.
given a delta t or a change in t we can find the delta t' for any reference frame by multiplying delta t by the quotient of the reference frames rate and earths rate. Inversely given the passage of time in a given reference frame we can find the passage of time on earth with the equation ,
delta(t) = delta(t')*(earths rate / alternate reference rate)
meaning that if you experience 1 second traveling close to the speed of light, with a rate of experience 0.001 hz on earth 1 * 1/0.001 seconds have passed.
Now you start at t=0 at earth, you wait 10 minutes then go to some planet P. while in transit you experience 1 minutes of experience.
Now at the planet you stay for 5 minutes, dropping off a friend.
Next you return to earth, again experiencing 1 minute.
So when you return home t = (10 + 1/s + 5/p + 1/s)minutes
Meanwhile a clock in your pocket reads 17 minutes. Knowing that s < 1 and assuming p is also, t > 17.
At that same time the another clock t2 left with your friend at P reads: t2 = (10 + 1/s + 5/p + (1/s)/p) m
where the (1/s)/p represents 1 minute of time passing on the spaceship worth of time passing on P.
I hope this explanation helps you wrap your head the idea!
** I am not a practicing physicist take my words with appropriate skepticism. **