I'm no physicist but (like Oniya) I've read some about this over the years; also for what little worth it has, I know some people who have been doing research in astrophysics and have discussed this stuff with them in a relaxed way.
First off, 600 mph is nowhere near the kind of speed where you would begin to have relativistic changes of speed from different angles; 600 miles per second wouldn't be of any consequence either. And as long as the grenade is in the atmosphere, or building up to escape velocity (about 7miles/second from earth - let's suppose the planet in the example has this EV) it doesn't pick up the planet's orbit speed as any part of its own speed from the point of view of someone observing it from the planet. If it were observed from another planet much further outside, yes, then it would be seen to travel with that speed plus its own speed when it was orbiting "forwards" and then v=planetary orbit speed minus object's own moment speed, when it sort of went "around the bend" into the other half of its orbit.)
If it broke free completely from the gravity of the planet it had been sent from, and went into a path taking it away from it - like a probe for Mars or the outer planets - it would keep the speed and orbital direction it had at the point when it completely broke off the "home planet gravity field" which is *not* the same as the point where it enters into orbit around that planet: any satellite has to reach escape velocity to enter into orbit, but if it's lost in space later, near earth, or between the earth and the moon, it will still slip into orbit around the earth for many years. So it's only when they have gone some way beyond the distance to the moon (in earth's case) that these probes or whatever it is escape any substantial bending power from the earth on their paths through space.
Once it has got that far, and starts travelling freely through interplanetary space, if it has some engines of its own turned on during the entire trip to let's say the next planet, it won't pick up the home planet's orbit speed in any simple way, no matter from where it is being observed. A shock wave erupting from a point hat far out would be, I think, the "shockwave speed" added to the speed of the grenade just before it exploded, and theoretically it would travel back to its home planet, or on to another planet with that speed, if those planets were moving at a slower speed or moving towards the spreading circle of the shock wave, meeting it (actually I think any sizable planet in the inner parts of a planetary system, even Saturn in our own neighbourhood, would be moving much faster through its orbit than most probes or grenades could hope to travel - so they would often outrun the shockwave unless it happened "in front of" said planet). .
The trouble is, most probes we send out don't run on their own engines all the way out. The engines are shut off as soon as the ship or probe has been circling around earth a bit and then pushed itself off into an orbit aimed at the goal it is set to reach in a year or two. Loading Pioneer, Mariner or Cassini with fuel for travelling on their own power all the way through the solar system would be prohibitively expensive, even if it was nuclear fuel. With chemical rocket fuel it would be out of the question (the jury is still out on whether it could be practical to power the first manned trips to Mars with any kind of standard rocket fuel for the full length of the trip, though it owuld make the running time considerably briefer because you could just go the shortest way, pretty much a straight line). Once they have got up to a good level they just coast on the orbits they will get into through the gravity of the sun and the planets nearby, and the speed in that orbit is not a simple product of neither the home planet's orbital speed, nor the vessel's own maximum speed as it took off from the planet.