Per present consensus we can't exceed the constant c, but if we could accelerate to 1% that speed and slow down again when desired then amazing things become possible. Colonizing the galaxy would be a slow process at that speed, but if humanity's survival is no longer centered around one planet then we have plenty of time.

That would mean not just putting human boots on Mars, but extensive exploitation of the solar system. Mine Mercury siphon Venus forgeworld Mars, siphon gas giants to power the 'slow' interstellar ships.

Pick 4ish nearby star systems and send slowships (generation ship, longevity ship, cryo sleep ship whatever we get a firm grasp on first) one after another in slow processions. Orbital colony networks around the big cloudy worlds assemble and fuel up the slowships to be completed every year or every 10 years or every 40 years whatever. Each big cloudy world gets one target star system to attempt to colonize.

First slowship seeds a star system with comm sats in star orbit, second slowship deploys smaller drones to put scanning sats into polar orbits of planets, next several slowships transit space station parts and builder bots into the system, then we send human pioneers then we send colonists. Once motherships are done transiting to the star system they can be repurposed as giant communication devices.

This is with conservative expectations of technology but it involves a little bit of faith in humanity.