We will be sending entire civilizations of nanobots, or at least small artificial organisms.

At 5% of the speed of light, including travel, reproduction & relaunches, a civilization like that at the center of the galaxy will reach the rim in 100,000 years.

At the nano level time effective massively expands. Activity at that scale operates with speeds many magnitudes faster than gargantuan Ent-like beings like us, can easily imagine. :)

It isn’t a journey suitable for humans, and our position here is changing rapidly decade to decade. The galaxy was never going to belong to humans, but hopefully our progeny.

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Then there is hope from physicists, that harnessing stellar level energy will allow faster than light warp travel. That would let beings cross the vast spaces between galaxies.

Not at all depending on the timeframe you want to talk about in.

For example, one way to travel around the galaxy is to find some other object with a very fast orbit and colonize it. Then as it goes along you throw off seed ships. This is the whole idea behind 'von neumann probes'. Massive numbers of things working in parallel can get a lot of work done.

But were you saying "get started on this in 1000 years" instead of finish in 1000 years?