What is the relative velocity between planets in different solar systems?
Tens of kilometres per second for nearby stars. Hundreds of km/s for stars in distant parts of the galaxy to thousands of km/s for stars in nearby galaxies. Anything from "much more" to "it's complicated" for stars in very distant galaxies.

The Earth is orbiting the sun at 30km/s and the orbital velocity of exoplanets is on the same order of magnitude. The stars are also moving relative to the sun. We can measure the transverse velocity for nearby stars by seeing them slowly change position (relative to the distant background stars) and we can measure the radial velocity by the blue- or red-shifting of the light. 

We can't directly measure the transverse velocity of distant stars, but their general motion is around the galaxy with a velocity, relative to the sun, in the 100s of km/s

Nearby galaxies are moving at similar velocities. Again, we can currently only measure the radial velocity, but their transverse velocities would be of the same order of magnitude, and in the 100s to 1000s of km/s.

Distant galaxies are part of the Hubble flow, and the expansion of space, so defining relative velocity or relative distance becomes tricky and needs to be done in the context of General Relativity, at which point things become more problematic, and beyond what is required for space opera.

For context, spacecraft in orbit around the Earth travel at less than 8 km/s and our fastest spacecraft (which are needed to travel close to the sun) have never reached 30km/s.