Assuming that humans will eventually be able to build spacecraft that can accelerate to speeds of 0.001 to 0.1 of the speed of light, and decelerate at the end of their voyages, it should take those spacecraft about 10 to 1,000 years to travel one light year.

Assuming that humans can colonize star systems without habitable worlds by building space habitats, and that the average star has several star systems with 10 light years, and a voyage of 10 light years taking 100 to 10,000 years is the maximum length, a world will colonize all the star systems with 10 light years within 100 to 10,000 years of sending the first colony ships.

Assuming that it takes 100 to 1,000 years for each colony system to become populated and adanced enough to send out colony ships of their own, a world should colonize all the stars within 10 light years within about 200 to 11,000 years after being colonized.

Thus humans would be able to spread across the galaxy at the rate of 10 light years every 200 to 11,000 years, at an average speed of 0.000909 to 0.05 times the speed of light, and thus spreading at an average rate of 0.000909 to 0.05 light years per year.

Since the farthest parts of the galactic disc are probably much less than 100,000 light years from Earth. at that rate it should take less than 2,000,000 to 110,000,110 years to colonize the entire galactic disc of the galaxy.

But it would be impossible to colonize those parts of the galactic halo where the density of stars is low enough that most stars are more than 10 light years from their nearest neighbors. Thus it would be impossible to reach most or all of the globular star clusters and most or all of the small satellite galaxies of the Milky Way Galaxy, let alone more distant galaxies in the local group like the Andromeda galaxy, let alone galaxies tens or hundreds of millions of light years away, let alone galaxies billions of light years away.

But if humans are able to built spaceraft which can travel much faster than 0.001 to 0.1 times the speed of light, and/or are able to build spacecraft capable of deliving human colonists to distant stars after voyages much longer than 100 to 10,000 years, humans could spread much faster and reach much more distant locations within a single voyage.

Thus humans might be able to spread from star to star even among the widely separated stars in the halo of the Milky way Galaxy, and thus reach globular clusters and even the small satellite galaxies.

And if humans develop spacecraft which can travel at almost the speed of light, so that time dilation slows down time aboard the ships to a tiny fraction of the passage of time in the outside universe, a ship could travel 100,000 or 1,000,000 light years in only a few decades of ship's time, and humans could conceivably spread from galaxy to galaxy.

And if it every becomes possible to build ships with Alcubierre drive or some other form of faster than light travel, the sky will be the limit, as the saying goes.

Part One of Four:

Assuming relatively realistic and probable space travel.

Assuming that humans will eventually be able to build spacecraft that can accelerate to speeds of 0.001 to 0.1 of the speed of light, and decelerate at the end of their voyages, it should take those spacecraft about 10 to 1,000 years to travel one light year.

Assuming that humans can colonize star systems without habitable worlds by building space habitats, and that the average star has several star systems with 10 light years, and a voyage of 10 light years taking 100 to 10,000 years is the maximum length, a world will colonize all the star systems with 10 light years within 100 to 10,000 years of sending the first colony ships.

Assuming that it takes 100 to 1,000 years for each colony system to become populated and adanced enough to send out colony ships of their own, a world should colonize all the stars within 10 light years within about 200 to 11,000 years after being colonized.

Thus humans would be able to spread across the galaxy at the rate of 10 light years every 200 to 11,000 years, at an average speed of 0.000909 to 0.05 times the speed of light, and thus spreading at an average rate of 0.000909 to 0.05 light years per year.

Since the farthest parts of the galactic disc are probably much less than 100,000 light years from Earth. at that rate it should take less than 2,000,000 to 110,000,110 years to colonize the entire galactic disc of the galaxy.

But it would be impossible to colonize those parts of the galactic halo where the density of stars is low enough that most stars are more than 10 light years from their nearest neighbors. Thus it would be impossible to reach most or all of the globular star clusters and most or all of the small satellite galaxies of the Milky Way Galaxy, let alone more distant galaxies in the local group like the Andromeda galaxy, let alone galaxies tens or hundreds of millions of light years away, let alone galaxies billions of light years away.

Part Two:

Assuming much faster and less probable space travel.

But if humans are able to built spaceraft which can travel much faster than 0.001 to 0.1 times the speed of light, and/or are able to build spacecraft capable of deliving human colonists to distant stars after voyages much longer than 100 to 10,000 years, humans could spread much faster and reach much more distant locations within a single voyage.

Thus humans might be able to spread from star to star even among the widely separated stars in the halo of the Milky way Galaxy, and thus reach globular clusters and even the small satellite galaxies.

Part Three:

Assuming the space travel capabilities in the original question.

And if humans develop spacecraft which can travel at almost the speed of light, so that time dilation slows down time aboard the ships to a tiny fraction of the passage of time in the outside universe, a ship could travel 100,000 or 1,000,000 light years in only a few decades of ship's time, and humans could conceivably spread from galaxy to galaxy. Thus humans could conveivablye reach every galaxy which is not so far away that it is receding faster than light, or faster than the very close approximation to the speed of light which such ships could achieve.

So humans could probable colonize every galaxy that is now no more than a few tens of billions of light years away.

Part Four:

Assuming FTL

And if it ever becomes possible to build ships with Alcubierre drive or some other form of faster than light travel, the sky will be the limit, as the saying goes.