why did they search for a new world

Exploration. Adventure. Reduce the overpopulating of Earth. That's probably about it.

Curiosity, resources and the urge for humans to destroy I guess. kinda like Star Trek

Maybe people wanted to start fresh, be somewhere that's more peaceful I guess

I agree on curiosity. I mean after the events of the Space Finale the whole planet was pretty much aware of who the Power Rangers were, and were likely made aware of the fact that some of the Rangers were from other planets.

So it must have got them thinking that if those aliens are willing to step up for Earth there's no telling how many other allies can be made to keep humanity safe for the rest of eternity. So why not use their help to find other planets for future generations to be able to live on?