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My question is as follows - let's assume our objective function is to have as many humans as possible living outside the confines of the Earth as soon as possible. For this, we need to provide those humans with some habitat outside Earth. One approach is to try and colonize other bodies in the solar system like Mars. Another is to build our own habitats, starting with the vicinity of Earth. Which of the two approaches will get us further in terms of the stated objective function if we start investing resources into one or the other now?

There is talk of inhabiting Mars and making life multi-planetary. I personally think the Moon is a better bet, but another approach is to simply start expanding the International Space Station so it can support more and more people. Then, we can simply build more of those. These large ships would start to collect more and more energy from the Sun, which is going towards a Dyson swarm. The question is, why even bother going to Mars and inhabiting it as opposed to just disassembling it for material when the time comes? Is it substantially easier to build a life-sustaining habitat on Mars than an independent spaceship?

The advantages Mars might have are that it has a lot of rock under which a human colony can get shelter from incoming radiation. Also, it has some gravity which we humans are accustomed to (though just a third of Earth).

The advantages of the space-station approach is that you can stay close to Earth. This is valuable in terms of being able to get supplies there much faster, help arrives quickly in case of emergencies, the close contact ensures much less psychological hardship for the inhabitants who can always go back to the planet for visits, do Skype calls with relatives, etc. You can also stay within the Van-Allen's belt so you don't need too much extra shielding, at least in the initial phases. Also, for intermediate economic motivation, this ever-expanding space station could start as being an exotic hotel for the wealthy.

So, if our goal is to have our species acquire a substantial footprint outside the Earth, should we invest our resources in going to Mars or the Moon and set up shop there or simply start expanding the space station to the point of a city and then rinse and repeat? To me, the latter option seems much more logical but am interested in alternate opinions.

Mars is the better bet

There's a reason why building habitats on Mars makes more sense, and many of them are covered in the Mars Trilogy by Kim Stanley Robinson. The simplest and most plausible reason though is that Mars is already a 'spaceship' in a stable orbit around the sun, so all you have to do is populate it with the people and resources you need. You don't need to design spinning habitats, then build them while depleting the Earth's resources to do so. In theory at least, Mars is likely to have some mineral wealth that can be exploited once we're there and there seems to be some compelling evidence that it also has a water (or ice) table that can be exploited as well; for drinking, fuel and oxygen.

Whether or not Mars could be terraformed in the same manner as described in the trilogy above without massive investment from Earth is a question for which I don't have an immediate answer but if it can, then it's going to beat out habitats every time simply by virtue of the scale. You may be able to put thousands on a habitat, but Mars could (potentially) hold up to a billion. That is likely the upper bound I grant you, but the simple reality is that in terms of scaleability Mars will beat habitats every time.

If you're looking for the greatest possible numbers off the earth in the shortest possible time, and you've got unlimited resources to do it, then Mars is a bit like a steam engine car racing a petrol engine car; if you're measuring shortest possible time in terms of years, or even decades, then go with the habitats because Mars simply can't scale up that fast. But, if you're measuring shortest possible time on a scale of centuries, then go to Mars straight off the bat. It'll take longer to ramp up, but once you get it going you'll simply be able to house more people faster because building a habitat on the surface of Mars is going to be MUCH simpler than building one on Earth, launching it into orbit and assembling it, then filling it with air and populating it.

Also, there's a chance that Mars might eventually be able to build habitats out of materials sourced from Mars, whereas that will never happen in orbit. That means that your efficiency goes through the roof.

A quick comment on the Moon; pretty much everything that I've just said about Mars could also apply to the moon, except that it's unlikely to have water reserves. Also, we don't know a lot about mineral wealth housed on the Moon, although He₃ has to be a great energy source if we can use it in fusion generators. The moon would also have the advantage of being closer, and the lower gravity well by comparison to Mars also means that if the moon ever has anything to export, it's going to be able to do so at a much cheaper rate than Mars can.

All that said, if people have to be off the Earth, whatever is going to cause them harm there, if it's astronomical in nature (say the sun, or an asteroid) is just as likely to cause issues for the Moon as well. Also, no water would mean that lots of it would have to be shipped up (at great cost) with potential colonists, and being smaller means that the upper bound of colonists is much smaller than what might be possible on Mars.

But again if you're not staying there for the long haul (no pun intended) and you're only interested in numbers over a short period, it might be a viable option, especially with the aforementioned unlimited resources to do it.

First, we don't know how to build a self-sustaining ecosystem that will support humans indefinitely at anything much smaller than planetary scale. Consider that the ISS needs a resupply mission every month or so, just to support a crew of 3-6 people.

Second, the difference between a (terraformed) planet and a Dyson swarm is like the difference between a country estate and an urban tenement. The quality of life would be vastly better, and since people who can afford to go to space on their own dime (rather than on missions paid for by governments) are going to be wealthy, the amenities matter.

PS: Of course we could envision situations, such as Heinlein's "The Moon is a Harsh Mistress", where the Dyson sphere or unimproved planet/moon is treated as a sort of super Sibera/Australia, so only "undesireables" are exiled there.