Distance between Earth and Jupiter is not the same all the time. Because of their elliptic orbits around the Sun the distance varies. The smallest interval for light to travel between the two is 32 minutes and the longest is 53 minutes. Also Jupiter complete an orbit in 11.86 Earth years.
If a ship left a planet (in this case Callisto, a satellite of Jupiter) to travel to a known destination (in this case, Mars) and you were trying to ascertain where that ship was to reach it, how would that be done? Would the ship be able to be picked up by telescopes, or would you track the possible courses, or would there be a *ping* on the ship - it is of Earth origin? Which are likely and which impossible!
I believe the most used should be based on the known itinerary. A ship should have some sort of a flight manifest (like today planes have) where the course and speeds are declared. So based on that manifest one could predict the approximate location of the ship and send the communication signal in that region.
The ping is also a good idea, but it should be completed by space beacons. Also a ping can be assimilated to the transponder a plane use.
There is also the radar which can give you the position of the body in space.
You should read about the methods used for plane tracking in modern military warfare. Humans are lazy and they will not use completely different techniques once in space.
Put a spy on that ship who activate a tracker. Or a gift from a spy which is a tracker. Or a computer virus who make the ship give up its location without the crew knowing it.
And voila, it will save you a lot of tech details and time needed to document.
There are going to be known possible courses. On a bigger scale, there is the Interplanetary Transport Network, using gravity assist and Lagrange Points to move between different planets with the minimum of fuel. The theory for this navigation already exists.
Telescopes can be used, to track the reflection of the ship. As it moves, you would be able to predict it's route, within reason. Straight (well, curved with gravity) lines are going to be the norm, as changes in direction cost fuel, and that's a finite resource.
Transponders are good for tracking. Over that distance, the signal is only going to be few minutes old, but is subject to interference -- dust, rocks, planetary bodies. Best thing would be to have multiple receiving stations, e.g. one on the moon, one in the Sun-Earth Lagrange point, two or three on Earth, another couple on Mars. They could all be dedicated to tracking craft, similar to radio telescopes working in concert.
The 'ping' is the most subject to failure, as it's a machine. Transponders do fail. Comms cut out. Albedo reflection is variable but, once you know what to look for, you can find it again (that said, astronomers have lost a few asteroids over the years, so...). A ship's course between A and B may have variables, but alter it too much, and you go off Into The Great Wide Open. Of course, if those tracking the ship did lose sight of it for a long period of time, they might believe that was what had happened.
a lot of this depends on your drive system, how much acceleration you have in hand, reaction mass - and of course the geographical (astrographical?) position of your moon, Jupiter and Mars. Most of today's craft rely on maximum thrust for short periods of time and ballistic (freewheeling) for almost all the trip (plus slingshot manoeuvres and the like. But they don't tend to use much drive). Now, if you see the craft leaving Jupiter's orbit you can predict its path fairly accurately, and then you can track it with reflected sunlight, radar (if you've got a big, powerful enough radar dish in roughly the right place aimed in the right direction, lidar… but if you don't see it leaving, spotting something that small if you haven't got its original vector is close to being impossible - you might pick up course adjustments, or the last deceleration into Mars orbit, or the atmospheric braking to Mars' surface.
Next, go for a low thrust, continuous drive. Slightly easier to detect than a freefall flight, but not much. Not as easy to predict as the smoothly changing orbit, but at least some energy emission, so something to look out for. Still, that ship couldn't land on Mars - it'd have to stay in orbit and carry a landing craft, and it'll take a while slowing down.
What else? How about an Orion drive? A bit bumpy - letting off hydrogen bombs right behind your backside isn't all that subtle, and it's for sure the explosions would be quite visible to anyone looking in the right direction. Not very predictable, even for the pilot, but lots of waste energy which would take a while to dissipate. If you want a system you can easily detect at a distance, that's probably the best - the ship would be enormous as a ship (tiny as a celestial body) to damp out the shock waves, so would probably go for a large crew.