The Mars rocks we have on Earth were found here, especially in the Antarctic. They were blasted of the Martian surface by meteorite impacts and wandered to Earth in the following millions of years.
There are plans to send probes to Mars with a load of liquid hydrogen and a plant to convert that hydrogen and atmospheric carbon dioxide into liquid oxygen and liquid methane and send the rocket back to Earth with rock samples. The plant is to be powered by a Pu238 thermo-electric generator.
There is hope that on the moon as well as Mars we'll find caves that can be adapted . But we will also need earth moving equipment. The weight of rock above the cave need to be sufficient to counteract the air pressure within. Assuming we choose to have half of our sea level atmospheric pressure, a rock mass per cubic meter of 2.5 tons and taking account of the local gravity we need about 16 meters on Mars, 25 meters on the moon between the top of the cave and the surface of the planet to avoid inducing tension in the rocks.
Mars is somewhat drier than a desert, but the atmosphere still contains collectable water and lots of water is available near the poles. So let the first expedition land during spring near a pole and a cave, set up an energy system and start building the heavy industry you need. When colonizing on Earth you could start felling trees to get building material and fire wood. On Mars everything is more difficult.
Note - Your heavy industry needs ming for which you also need earth moving equipment. Send it in advance by slow Solar Sail.
While the points you made are generally sound, I do have a few concerns:
1. The human civilization has barely scratched the surface of the moon and Mars. Which means we know very little about the soil composition and what lies beneath. This means uncertainty in exactly what kind of heavy machinery is suitable for the conditions on the moon and Mars.
2. Sending the heavy equipment first makes sense and has been proposed by many sci-fi writers for many years (decades even). The biggest hurdle is cost. Building a space station in earth's orbit already requires numerous countries to share the costs. The cost of sending a manned expedition to Mars is going to be a lot more expensive than that of the International Space Station.
3. There is a lot of speculation on exactly how much water there is on Mars. Before trying to establish a permanent manned presence there, it would be necessary to determine with greater certainty exactly how much water there is in the atmosphere there, and beneath the surface.
4. The time taken to travel from Earth to Mars is quite considerable (~9 months). Feeding the astronauts during that trip itself is already quite an interesting challenge. Than, there's the issue of the effects of living in zero gravity for that duration (such as loss of bone mass). Currently, after a space trip, astronauts go into medical treatment for these issues. That's hardly likely to happen after they land on Mars though, unless medical personnel follow along and medical facilities are made available on Mars as well. That again drives up costs.
While I have no doubt that the technological challenges can be solved, its the financing and the political will to commit to such a huge endeavor that I'm not so sure about.