I find that depending on your start you have to adjust what you do.  For instance getting the free colony ship or 3 constructors is pretty game changing early on (from ideologies).

Also I find that other factions will sell you their colony ships.  This is being fixed I believe but it is pretty exploitable.  Also you start with 5k....what are you buying / rushing with it?  I usually rush 2 colony ships.  Which helps colonize things early.  Also what size map are you playing on and with how many opponents?  If you play on a tiny map and have to many opponents than all the stars are just their home systems which means there isn't a lot of things to colonize.

There are other things you can do to help yourself out.  For instance the 3 free constructors you can upgrade to colony ships for less than 100 credits each give you 3 colony ships (don't know if this exploit will be fixed eventually).  The only down side is you need to load them in order to colonize.  

You can also give yourself the ability as a custom race to colonize some extreme worlds from the beginning (aquatic, frozen and baron) worlds.  This will allow you to colonize some worlds you couldn't otherwise giving you a little more of an advantage.

As Vorell stated you could trade for colony ships, technologies, trade resources, or other items to help give you the advantage.

There are many ways to give yourself a quick start besides starting location.  Eventually Stardock will have more arena maps that will be pre-built maps giving every player more of a equal starting location.

In that case I suggest changing your settings....add more stars / planets / inhabitable planets...and you will get a more desired outcome.

I avoid maps smaller than medium and usually go loose clusters and loads of planets to avoid the problem of only finding a few habitable worlds before the first rush is over. I love playing a custom Terran race with Knowledgeable as then I can see how close the other majors are to me at the start. If I do not have some room to expand, then I start a new game. I often send out cheap colony ships with only 1 billion pop so I can generate more colony ships in the early rush. To me it is more important to lock down more planets even with smaller starting pops rather than get fewer planets but each has 2 or 3 billion pop at start. I do not like to strip my homeworld pop down too low. Just a few ideas for your consideration.

   Just checking, you used constructors to extend your colonize range, made custom colony ships with more engines / range, extra scouts?

   Bad luck at game start can certainly be killer.