A lot depends on the size of your game. I play gigantic maps and in this case it's not necessarily how fast you explore but the rate at which you can produce colony ships that counts.
In all cases I simply use my colony ships as scouts and use my flagship to get money from anomalies.
In a smaller galaxy, I think most people agree that it's best to research up to impulse drive ASAP, but you still should be sending colony ships out as fast as possible.
In a large galaxy, I think it's a mistake to outright buy much of anything. You need that initial 5000 bc to last as long as possible because it's inevitable that you'll be deficit spending for quite some time. I do buy my first factory and then build my manufacturing capital on my home planet followed by building another 3 or 4 factories dependent on bonus tiles. My goal is to produce a colony ship every other turn (or even a bit faster).
You need to keep your approval at 100% to keep your pop growth able to sustain producing that many colony ships. I also usually take the 70% pop growth racial ability to support this. With all this I'm able to produce a colony ship every other turn and fill them with 500 colonists each.
If you can't keep your approval at 100% or you don't have the pop growth to support 500 colonists every other turn, reduce it to what you can support, generally at least 300 is almost always possible.
If you do this you'll outcolonize the AI at any level, which puts you in a far better position than if you don't.
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