I know that rare earth metals aren't too rare and the name is mainly due to their low density. I also know that phosphorous is comparatively rare but is hyper concentrated on Earth due to its importance in biology.
So outside of Earth, how abundant would these resources be? Where would you best mine them? In asteroids, moons, rocky planets, or gas giants?
Any star similar to the Sun should have at least a couple planets like Earth. Even if they don't have a biosphere, the amount of materials in the planet should be about the same. Just like Earth and Venus are very different, but if you could put both planets in a centrifugue and liquify them, they would become very much alike.
G Dwarfs are about 7% of the stars in the Milky Way. So if you ignore all other stars you still have way more than enough. Just fly around the galaxy at around the same distance from the center as the Sun is, +-20% that distance. Statistically 7% of the stars you find will have planets with lots of phosphorous.
phosphorous is comparatively rare but is hyper concentrated on Earth due to its importance in biology.
The universe appears to be about 7 parts per million of phosphorous (source). It isn't precisely commonplace, but it is far from rare. It is more common than copper, for example. It may be hyperconcentrated on Earth in biological things, but by the standards of modern technological societies, it is far more economical to exploit mineral sources of the stuff. This will likely also be true for your spacefarers... leave the guano for the pre- and peri-technological societies, because they need less of the stuff.
So outside of Earth, how abundant would these resources be? Where would you best mine them?
Rocky planets with a metallic core around Sunlike stars at similar distances from the galactic center would obviously be a good place to start looking.
You'll find the bulk of Earth's stock of the stuff to be quite deep under the crust and therefore awkward to harvest. Smaller astronomical bodies with a common origin would therefore be much easier to mine. Earth's moon is a particularly special case, as it probably formed from a big impact which stripped off a big chunk of the Earth and as a result it formed vast KREEP terranes where KREEP stands for "Potassium (K), Rare Earth Elements, Phosphorous".
In asteroids, moons, rocky planets, or gas giants?
Forget gas giants. Gravity wells much too deep. it might be practical to extract some kinds of useful light element (eg. helium-3) from a gas-dwarf or Neptunian, but the things you're after are too heavy to form enough of the atmosphere to be practical to exploit and the rocky core is much too hard to access.
Dense metallic asteroids would be a favorite because it takes so much less energy to completely dismantle the things compared to whole planets. You can probably find a reasonable amount of stuff like neodymium and other "rare earth" out there, and though the densities are low they'll still be practical to exploit when you extract the more common metals and minerals.
Lanthanides have been suggested to be more common on Earth than in asteroids, so really your best bet is to strip mine Earthlike worlds and their moons. If you can afford to fly between stars looking for metals, then you can definitely afford vast solar-powered mining facilities that can do the concentration economically for you.
Your ideal target system, perhaps, would be one that had large asteroid belts in Earthlike orbits around Sunlike stars, because there's a good chance they'd have a similar mineral composition to the Earth but without that inconveniently deep gravity well or thick, hard-to-mine mantle. A late-stage protoplanetary disk might also be a good prospect. Neither location is likely to have locals who evolved their naturally, by way of a bonus.
You would mine phosphorus where it is hyperconcentrated.
Depicted: The Burns Omni Net. Mr. Burns uses it to sweep the ocean clean of animals, from which he makes Little Lisa Slurry.
Your aliens desirous of phosphorus would harvest the bounty of life bearing planets: their life forms, each a rich concentration of phosphorus. A human body contains 1% phosphorus! You are not going to find that in some extraterrestrial ore!
Ok, guano. Guano deposits have a lot of phosphorus. So life forms, and guano.
"So outside of Earth, how abundant would these resources be? Where would you best mine them? In asteroids, moons, rocky planets, or gas giants?"
You are asking three different questions with the first being very different from the 2nd and 3rd.
In reply to questions #2 and #3:
Anywhere
Frame Challenge
An interstellar civilization would know how to obtain energy from matter and matter from energy. E=mc^2.
Thus you can have matter from matter adding the necessary matter/energy.
Of course this is more easily said than done.
And yet, we know it's possible. That is how all the elements of the Universe formed save for the 5 lightest ones (generated early in the aftermath of the Big Bang).
Later in the twentieth century the transmutation of elements within stars was elaborated, accounting for the relative abundance of heavier elements in the universe. Save for the first five elements, which were produced in the Big Bang and other cosmic ray processes, stellar nucleosynthesis accounted for the abundance of all elements heavier than boron. In their 1957 paper Synthesis of the Elements in Stars, William Alfred Fowler, Margaret Burbidge, Geoffrey Burbidge, and Fred Hoyle explained how the abundances of essentially all but the lightest chemical elements could be explained by the process of nucleosynthesis in stars.
Nowadays, with our limited understanding and technology, we can already achieve artificial transmutation through machinery that has enough energy to cause changes in the nuclear structure of the elements. Such machines include particle accelerators and tokamak reactors.
Clearly we can only do it in a very limited way and with costs which would be unacceptable compared to the value of the element.
But things may well differ for an interstallar civilization.
The very reactors that propel their ships among the stars may also generate huge quantities of valuable rare earth elements as a by product.
