[Phantasm]

What I did in mine followed the following steps:

1) First extrasolar colony took 20 years to reach at fast STL speeds.
2) First contact with another species that already had lightspeed drives.
3) War breaks out between the species, fought mainly at STL and lightspeed for 150 years.
4) FTL drives developed not long after the war ends.
5) Sleeper ships sent out to many stars at fast STL, lightspeed, and slow FTL speeds. Some ships still unaccounted for.
6) Extrasolar colonies founded on nearby systems not long after.
7) Cloning used in a lot of areas to quickly populate some planets, generally used as a workforce on farms, in mines, and in construction.
8) Colonization of a 20 ly radius assisted by well-mapped routes. The frontier extends out to 50 ly beyond that, not so well mapped and often in very small populations.

Mind, my setting takes about 500 years to go from TL8 to TL11^; the setting itself is just entering TL11^ ca. 2500 CE.

No world other than Earth and some alien homeworlds has over a billion people; even the leaders of my "self-determination league" measure their populations in the millions.