It's a hell of a long way to another galaxy, even at warp speed. The trip was attempted once in "By Any Other Name" when the Enterprise was hijacked and outfitted with an improved drive to get to Andromeda. Even then, the trip was going to take over a century.

It is worth noting that normal warp drive does not even suffice for trips between galactic quadrants. The Alpha Quadrant is the normal beat for Star Trek episodes. The Gamma Quadrant can only be reached via the wormhole near Deep Space 9. This wormhole is apparently only stable because it is inhabited. The starship Voyager got itself hurled into the Delta Quadrant, and faced a decades-long trip home. Fortunately, a shortcut was found.

God only knows what's in the Beta Quadrant. Nobody seems to care.