The galaxy is about 100,000 light years across. The speed of light doesn't need to be exceeded to allow for a civilization to spread out and occupy the entire galaxy in 10 million years from GO! Or even 1 million years. Getting to the next galaxy [Andromeda] is a different story.
That may not be a problem a couple of billion years from now. Our galaxies are on a collision course. In 2-3 billion years the MW and Andromeda will be one mega galaxy.
And the kids in the backseat asking if we are almost there for that amount of time is the reason we haven't seen any aliens yet.