The second part of this answer is a non-sequitur. Why Mars in particular? The technology needed to get there doesn't carry over to any other destinations except the Moon and Venus, and it's much easier to build next-generation rockets on Earth. Why not just skip Mars and go straight to step 2?
You need resources to build with. Venus isn't an option because you can't get resources from it. (If there's two things you need, it's energy, and physical materials.) The Moon is certainly an option, but while Mars is a desert, the moon is volcanic rock and never had active geology that concentrates metals. Additionally while Mars has an atmosphere of sorts, the Moon has none which means anything mounted on the surface of the moon is in an equivalent environment to being in deep space. This means there's zero protection (besides the planetary body itself) from radiation and micrometeorites (and bigger meteorites). They rain down constantly on the moon slowly carving divots into whatever you build things out of or punching holes in them.
Also I disagree that the technology to get to Mars or the Moon is inapplicable to elsewhere in the solar system. And most of the other options are generally worse than Mars for one reason or another. I'd like to hear what you think "Step 2" actually is.
We do! And for a long while most of the resources will come from Earth. However you can't keep doing that forever as the costs will grow to an unsustainable level.
If you could magic up a civilization on Mars today, it'd exponentially cheaper to launch materials from Mars to elsewhere in the solar system than to launch those same materials from Earth. Similarly it'd be cheaper to dig those materials out of the ground for use on Mars than it would be for them to be used directly on Mars.
Counterintuitively, it's cheaper to launch materials from Mars to Earth's Moon than it is to launch them from Earth's surface to the Moon.
One note on the "exponentiality" in the other comment. Getting from Earth's surface to the moon (without staging) requires a rocket that is approximately 99% fuel by mass (calculated using rocket equation and engine performance of a high quality engine). That leaves you with very little margin for building a rocket's structure which means it needs to be built extremely carefully with little room for error. Also in order to get reasonable payloads, you need to make rockets of tremendous sizes in order to get reasonably sized payloads to locations.
On the other hand a rocket that flies from Mars surface too Earth's moon only needs to be 90% fuel which is a lot easier to do with modern materials. This also means that you can carry significantly more payload as your total mass.
With some napkin math, this allows you to launch, with the same rocket, from the surface of Mars to Earth's moon a payload 10 times bigger than if you launched it from Earth. The penalty you pay getting stuff out of Earth's gravity well is just simply huge.
> The technology needed to get there doesn't carry over to any other destinations except the Moon and Venus.
The technology for living there, and some of the technology for going places, does. Would you really want to try to go to Epsilon Eridani without going to Mars first?
> it's much easier to build next-generation rockets on Earth
Ultimately probably not; Earth's gravity well is too deep. Mars is, AFAIK, the only place we could conceivably build a space elevator with current materials technology, so if we assume interstellar rockets will need to be built with materials from both asteroids and planets then Mars orbit is likely the best place for that.
>Mars is, AFAIK, the only place we could conceivably build a space elevator with current materials technology
That's not true: the Moon would be much easier.
But if you're just comparing Mars to Earth, then definitely yes. Building a space elevator on Earth is sci-fi for now. I'm not sure about Mars, but on Moon it would be rather easy actually, since not only is the gravity 1/2 of Mars', but there's no pesky atmosphere.
Absolutely true, but my point is that there's no destination beyond Mars reachable with the spaceships we build to get there. So I don't get how it's some sort of gateway rather than a detour / dead end.