I think that probably we won't in our present form. Conceivably might do so if we evolve and transform in some way to be more peaceful, wise (probably long lived also).
Reason being that creatures able to colonies the galaxy - just a few centuries advanced from us in technology - would have such awesome powers that they could easily let loose self replicating machines with the ability to reshape the entire galaxy - but with no way to communicate with each other in real time - and no way to act together to deal with hazards - not with the light speed delays - and no way to meet together to consult to deal with difficult situations.
And - if we continue as we are now - then amongst the hundreds of billions of trillions of colonists in the galaxy -there would be at least a few with - either just plain silly ideas such as to turn entire galaxy into paperclips - or people who are too clever by half, who think they have the solution to everyone else's problems in the galaxy - or dictators, or mass murderers, or beings with chimpanzee like intelligence who have inherited machines of vast power from their ancestors, etc etc.
I think any sensible ET contemplating such a future will decide not to colonize the galaxy.
The few who react by saying "Look, galaxy seems to be not colonized yet and up for grabs, let's go out and get it" - are so reckless and immature that they probably destroy themselves long before they have the ability to do it.
Some ETs may have the wisdom, and also ethical values, and clarity of vision and sympathy and understanding to do such a thing. Maybe even us in the future. But right now, if you mean humans in our present stage of development - and if you mean us with the power of self replicating machines able to reshape entire star systems and galaxies - I can't see it.