It's easy to find search results for how long it would take without FTL - but I'm wondering how long it would take with FTL, and there don't seem to be many estimates for that. Some very rough estimates I've seen are 100,000 years (Quora, SpaceBattles), since that's how long the Milky Way is, but then there are secondary concerns such as building colony ships and making the colony self-sustaining. I'm still working on the particulars of how FTL travel works in my world, but individual vessels can travel by warp speed and hyperlanes can be built that are capable of propelling ships faster. Space governance is possible with FTL travel and communication speeds.
I guess the speed of travel is somewhat arbitrary as it depends on the speed of the fictional technology - so exactly the speed of light is 100,000 years, 2X the speed of light is 50,000, etc. For reference, we can look at Star Trek speeds of travel. The United Federation of Planets had colonized a decent chunk of the Galaxy by the 2300's (SciFiStackExchange), and had accomplished this in around 220 years (UFStarfleet.org). The prediction by the Voyager to travel 70,000 light years back to Earth was 75 years (with stops), traveling at 9.975 warp which is around 1680 times light speed (SciFiStackExchange). Consequently, estimates for travelling from one side of the galaxy to the other in Star Trek seem to be in the 70-75 year range (YahooAnswers). By comparison, traveling across the galaxy in Star Wars, with it's ridiculously fast speeds (some estimates are at millions of times the speed of light), can be accomplished in days or only hours (Reddit). As a result, most of the galaxy in Star Wars has been colonized (SciFiStackExchange).
Colonization would of course begin pre-FTL within the solar system (like Mars, the Moon, space habitats, etc.). Some estimates state that it would take around 1,000 years to terraform Mars (Quora), so let's assume that once FTL is discovered ships are sent out to only the most hospitable worlds first, where terraforming is unnecessary or minimal. The question then becomes: how long does it take to colonize a new world?
The length of time to build a colony ship has been worked out before on this site as doable in less than 40 years (source). With an FTL-capable civilization, maybe we could bump that number even lower. The next secondary question is how long would it take to colonize a planet before it's self-sustaining - estimates are hard to come by, but this post says around 350 years (source). Perhaps this too would be reduced due to the speeds of FTL travel, since the Federation obviously did it in far less time. Neil DeGrasse Tyson also supposedly said colonization would be at an exponential pace due to colonies themselves colonizing other worlds eventually (SpaceBattles).
My reason for wanting to know this is I'm building a world where there are multiple alien cultures and factions (a la Star Trek), and I'm wondering how long it would take before one culture starts to have trouble colonizing because all the territory has already been claimed by another. There would be less conflict if there was plenty of space for everyone to share, but it would also be less interesting if anyone could get anywhere in the galaxy in the blink of an eye. I'm looking for a good medium, for the stake of drama, but also realism.
The numbers may be somewhat dependent on arbitrary fictional technology, so maybe it would be helpful to frame the question with known references: how long would it take for the Federation (Star Trek) to colonize the galaxy? How long would it take for the Empire (Star Wars) to colonize the galaxy? How long would it take us, if we discovered FTL at exactly 1x the speed of light, to colonize the galaxy?