4 colonies is easy.  Use the starting colony ship to settle a planet, take the Pragmatic choice, take the Ideology trait Builders to get three free Constructors, upgrade the constructors into cheap Colony ships you pre-designed, load up the new Colony ships and away you go.  How your Altarians did it I have no idea.

Nine ships sounds like god-like mode where they get free ships.  Or else someone bought a bunch of cheap scout ships.  Sometimes you get lucky with anomalies, but that would be a lot of anomalies.

Military ships require military hulls.  Look in the Exploration branch for the required techs.  The Weapons branch has things to put in those hulls.  You can hire Mercenaries with pre-weaponized ships while you are researching your own goodies.

It would be interesting to know, to which difficulty the AI was set.

4 colonies at turn 17 should however not be hard, even without the constructor trick. Just rush them (buy from your 5000 starting money) in your shipyard. Up to incredible difficulty I usually start with buying another scout ship, if I didn't find a planet so far a third one on turn two and if I still didn't find a new planet I buy vanilla colonizers, since I can explore reasonably with those as well. Otherwise I go for minimum colonizers, that means a self made template with only a colonization module.

Another thing you can do, is use the benevolent ideology tree, which gives a free colonizer, followed by a class 10 world and a class 16 at the end of the branch.

I should also mention that my custom civs almost allways have the +2 moves trait.

About the AI colonizing fast: The AI most likely just knows where all colonizable worlds are at start. Else I can't explain some of the incredible snatches they pulled on me early in the game.

Of course this all depends on your galaxy settings, too. You can change the habitable planet frequency for example if you would like to play with more colonies.

Hi. Here's my experience -- I hope it helps.

Galaxy Setup

I experienced a lot of frustration trying to find enough habitable planets in my early games.  I found the default settings made habitable planets too rare.  So...when setting up galaxy options for a new game, I would increase both "Planet Frequency" and "Habitable Planet Frequency" by one step until I found something I was happy with. 
I like Large galaxies, and so I settled on Planet Frequency=Abundant, Habitable Planet Frequency=Uncommon.  It's all personal preference and depends also on how many opponents you have.  For me and six opponents, this was good to have enough worlds to play with, but not so many that the game would become a management nightmare.  Here's my setup for a large galaxy with six opponents:  https://www.dropbox.com/s/9gqr8jbyj2rewyt/galciv3_large_galaxy_settings.jpg?dl=0

Altarians/AI

As for how the Altarians got four colonies....  Some races start with a single class 16 homeworld, but others, like the Altarians, get a smaller class 10 homeworld with a small class 6 neighbour that's easy to colonize quickly.  So that's two planets for them.  You can rush buy at least a couple of colonizers, and they probably got lucky finding nearby habitable planets (there is enough randomization that it's easier for some to get a quicker start than others).

I find the AI builds quite a few early scouts.  They're cheap to build and cheap to buy.  This gives them a larger number of ships, but they're completely harmless (no weapons).  More scouts means you can find habitable planets sooner, I guess.  I prefer to just build colonizers in the beginning, but I have to be extra careful not to run into pirates when hunting for new worlds (losing a scout to pirates isn't too bad, but loosing colonists to pirates really hurts).

Speed-Boosting Techs

The first two techs I always research are the early speed-boosting Engineering techs: Interstellar Travel and Hyperdrive Specialization.  These give all your ships +2 moves which really helps your colonizers and scouts get to those habitable planets quicker.

Ideology
erischild's advice above to get the first Pragmatic Ideology trait "Builders" and then upgrade the free constructors to colonizers is really good.  However, I usually retain one of the constructors to build my first economic starbase near my homeworld.