I think the answer of many people in this thread (buy colony ships) should really be clarified a bit, with the later patches and especially Dark Avatar.
I agree that in a galaxy setup that leaves just a few planets for each player, it might make sense to quickbuy colony ships. I like to play at big maps, however (usually huge, gigantic is a little too big for me), lately at crippling level. One thing that should be clearly understood is that using money to quickbuy things is vastly less efficient than using it to finance an economy that's running a deficit for a while.
So, my beginning of the game is as follows: I try to find a starting setup that gives me some manufacturing bonus on my homeworld (one +300% is perfectly fine) and then quickbuy the first factory on that tile, still in that starting round. After that, I just regularly produce colony ships. By the way I agree with the suggestion to use faster colonizers, because for one it gives you a better chance to win races to some planets, and also it reduces the time your population is crammed in the colony ships and not on some would being productive.
On your new worlds, I usually would not recommend buying anything, just start regular production on a starport and some factories, because even if you quickly obtain these two things, your new worlds won't have the population yet to make colonizers with 500M people on them.
While you are busy colonizing only from your homeworld, try to find some morale resources, research the first morale tech and lower taxes to try and get 100% approval on your planets. That is especially important for the new ones, so they start producing income more quickly.
If you meet other civs during the early time of following that advice, you'll notice that they usually have more planets, sometimes many more. However, since the AI is doing exactly what many in this forum suggested, quick-buying colony ships, they will soon go broke and need to raise taxes and reduce their spending level. During that time it is no problem at all (up to crippling at least, don't know about even higher difficulty levels) to overtake them in the colony race, and since you (should) have high approval at that time you'll soon have a lot more population than anyone else either.
During that phase of colonization, I typically run a deficit of 100-200 bc a round, so even if you are lucky enough to find some money with your flagship and the other survey ships you hopefully built, you'll eventually run out. For that reason, you need to build economy buildings fairly early, on your homeworld and on the new ones as soon as they have some real population. If you didn't wait too long with the economy buildings, by the time your cash runs out you can raise taxes and slightly reduce spending and your economy still won't be paralyzed, as it often seems to happen to the AI players.
At that point I usually have somewhat more planets than most computer players and a lot more population than any of them. My economy then is among the strongest too, which means I can at this stage happily go to war with one of the weaker opponents and consilidate my empire

For your race statistics, the importance of your economy rating simply cannot be overstressed. The +30% you can get with 4 points of your race picks are worth it almost always, and a federalist government can help a lot too (though I often pick the technologists instead). I usually play an evil race (the Yor) because the +100% economy rating from the mind control centre is just too sweet to pass by for me. I know MCC is pure cheese, but there you are.
The main point of my story: use your starting money wisely, quickbuying is not the answer because it's just terribly inefficient.