Asteroid mining isn't as obvious a win as I first thought. A cheap satellite costs a few million dollars. A ton of pure gold is worth $55 million dollars. You're going to have to collect many many tons of pure gold to make an asteroid mining operation worthwhile. How do you get 100 tons of gold from near-earth orbit to the surface of the earth? Split it into little chunks? Some heat shielding and parachutes per chunk, maybe? You've got to send those up ... is that going to cost more than the $55 million per ton of gold that you get back down?
I'm just an armchair physicist. There are plenty of more experienced people thinking this through, for example here, here, and here. My value add (if any) is if I stumble on some non-obvious solution.
Perhaps you could melt the gold in space and form some large hollow sort of thing. Or a big awkward gridwork. Or a large hollow sphere. If you can get enough air resistance so it falls to earth without melting or breaking up and without requiring supplies shipped up from earth, that would handle delivering the goods to the surface.
Moving the asteroid is done by shooting bits of the asteroid off in the opposite direction at as high a speed as possible.
I looked into one method, spinning a 1/4" steel rope 300m long so the end gets an acceleration of 10 earth gravities, and letting rocks go. That's about the breaking strength of steel. It has the advantage of mostly needing a rope and electric motor and solar cells, which are all reusable, and you can trickle feed the energy because a spinning rope in space will just keep going. It throws the rocks at 173m/s. Compare that to 6000m/s for rocket exhaust; it's a poor method. A tapered whip would be able to achieve higher speeds. Springs can also be slowly wound up.
A 1km-in-diameter asteroid weighs about 2 trillion kilograms. If you could shoot off a 1kg rock at 6000m/s, that changes its position by about .25 millimeters per day, or 2 centimeters over 3 months. A 100m-in-diameter asteroid is 1000x more maneuverable (25 centimeters per day per rock).
Chances are there aren't any gold asteroids. There's iron-nickel asteroids with trace concentration of gold (platinum iridium etc). There probably aren't even asteroids that are worth their weight in gold.
Spotting good asteroids doesn't sound hard. It might be enough to look at them. Orbiting them and looking at them tells you their specific gravity, which tells you how much metal they have.
Even if you need to sample them, it shouldn't be hard to make a reusable robot with springs that can land on an asteroid, drill a little, vaporize some samples, push off again, and gather enough solar power to rewind its springs.
It probably makes sense to have a team of robots, where some stay in orbit and are responsible for communication and power and don't have to worry about impacts, while another set lands on the asteroids, repairs things, and does other manual labor. More than one of each for redundancy, and something among them has to be able to repair broken bits, perhaps from a bag of spare parts, or by actually manufacturing parts.
Every look at a meteorite in a museum? Solid chunk of metal. What's it take to chip a bit of that off or process it? How are you going to build solar cells out of that raw material? Sounds hard.
But, it turns out, meteorites aren't representative of asteroids. Small asteroids are more of a rubble pile, with no or almost no strength holding it together other than gravity. The force of impacts ends up being dissipated in all the gravel, and it falls back together, which is why small asteroids still exist at all. They also have a lot more rock and carbon. There seems to be a cutoff at 150m diameters, where above it asteroids are rubble piles and below it they are solid hunks of metal/rock.
One implication of a rubble pile is all the heavy stuff is going to be rattled down to the middle over time. Small things (sand) rattle down to the middle too. The surface will be covered with light large rocks, because that's what floats to the surface.
Another implication is that you won't be drilling, so much as digging through gravel and preventing it from filling back into the part of the hole you've already dug.
If you're going to smelt an asteroid in space, either it's going to take a verrrry long time for a little robot, and it better be a super robust robot too, or you've got to manufacture the tools to do the smelting. You've got to build them out of the asteroid itself, which can be done with a little initial robot, as long as you can bootstrap up to large scale.
One low-tech approach is to build a concave mirror and concentrate sunlight to melt things. The surface of the sun is 6000 degrees celsius, and a very parabolic mirror could capture at least half of that, giving you 3000 degrees C to work with. That's enough to melt most metals (lead and gold and iron and nickel, but not tungsten). Water can be evaporated by unconcentrated sunlight (no liquid phase in space), yet it freezes in the shade.
Another approach is to make solar cells and use electrolysis.
Manufacturing solar cells in space gives us a concrete target. What's been manufactured in space so far? Not much. What's involved?
There are printable solar cells now. I see quotes of 1% to 20% efficiency, with the 20% cheating by using a monocrystalline substrate.
Printing a solar cell means first making a flat surface that doesn't melt when exposed to the sun, and accurately spraying it with various types of paint. You'd have to manufacture the paint too. There is carbon to be had on asteroids, so paint can be made (proof: wait, look over there, something shiny!). That leaves manufacturing a flat surface that won't melt.
If you have water, manufacturing any shape at all in space is pretty easy. It's solid in the shade and evaporates if you expose it to sunlight (so you need a bag to hold in the vapor while you're working, and you can de-ice everything by putting it in the sun). But if you try to make a solar cell with ice as the backing, the ice will evaporate pretty quickly. Lead is also very easy to melt and form. Iron less so, but it's still plausible. Some sort of 3D printer could make any shape from water or lead. Large flat surfaces ... you'd probably manufacture small flat strips and snap them together.
Can you add things to ice to make it stronger? Yes, see pykrete, though wood shavings won't be in plentiful in space. Various types of gravel and sand and dust will be plentiful though. And you will be able to manufacture more interesting types of fillers. Can ice be made strong enough for gears and other parts of machines, provided you keep them at -100 degrees C? Dunno. Wouldn't be surprised if you could.
Precisely laying down the paint on a flat surface is 3D printing technology again.
3D printing is overkill for flat surfaces. Melt metal and roll it through rollers. You have to get rollers, though. Probably 3D printing to make tools and equipment, then the equipment mass produces things. Ropes are wires which are again best produced by a dedicated process rather than 3D printed, but they need some initial equipment which might be 3D printed.