Gardening

As you already said, compost would be your top choice. You don't even need your own compost bin, as you could buy some from local composting facilities or neighbours. Or, you can start small with a worm bin (vermicomposting).

As far as drainage, using just rocks isn't the best idea imo. Perlite or LECA are great, but pumice for example is a bit more environmentally friendly if you don't reuse it.
They all have a thing in common: they create air pockets that prevent the soil from being waterlogged. Gravel makes the soil store less water, increasing aeration, but you need to water more often.

One completely different suggestion from my side would be to try !hydroponics@slrpnk.net That way, you don't need any soil at all :)

Horse manure (with or without straw/sawdust) is relatively easy to source (most horse places are happy to get rid of it) and can be mixed with wood chips (ratio ca. 50/50 just for a ball park value) for immediate planting. Most plants tolerate it well and it will turn into soil to be reused with more manure and wood chips in the future. Even if compost or more complex mixtures would be slightly better in theory this is absolutely worth doing for the immediate availability and ease. Two ingredients, and so far we've had sweet potatoes, beans, strawberries, tomatoes, turnips, carrots, beets, rhubarb, aromatic herbs, sunflowers, squash, cannabis and more

You can add wood ash (not too much, strong pH change!), charcoal (no worries, the soil loves it), organic kitchen waste (no need for a separate compost, just bury it in your soil mix!). But no need for any of this. You don't even need raised beds. I've grown in bags, large and small pots, in makeshift raised beds (metal grid and cardboard structure) and on no-dig soil (just dump on the soil, your soil life takes care of the rest).

I see too many people work in their garden with old-timey ideas of the hard work a garden needs and how it should look. Instead of sweating and getting frustrated you can always choose the shortest path, the fewest steps, the most uncomplicated solution. Never mind the aesthetics, it will look pretty once it grows. Don't forget to water!

Have seen people use hay bales to grow in, but you do have to condition them first. I guess it depends on your situation if that has to be a purchased material or not, but I am curious if the same idea can be used with other materials. I think the biggest "issue" with uncomposted organic materials is the nutrients are not as readily available, but I've heard that some things like squash love some uncomposted organic material to sink their roots into. I bet you can have success there, just maybe not the most optimal yields or anything like that.

Local coffee shops may have spent coffee grounds available for free.

You seem to already know compost is the answer - if you have the space it does not take very long to get a good pile going and start seeing some results.

But if you absolutely need soil now, check if there are any local community or industrial composing facilities nearby and call them to see if they offer fresh composed soil directly.

For drainage, leaves and sticks are the free option.

Also if it fits your space, you could consider swapping from hard pots to softer cloth pots which have great drainage and allow air to reach more of the roots.

For mixing in uncomposted food, it's easier to balance acidity if you have a bulk compost setup, but if you're selective about what goes in, chucking a few low-acid veggies in the bottom of your bins probably won't hurt anything. Hell, some things may even prefer acidic soils. Just bury them deep or critters will come searching.

Look around your area for composting facilities. In the US, they usually give it away for free, especially if you have a required community compost pickup program.

There may also have a quarry in the area that will let you grab refuse like shale or slab gravel you could use, but the consistency would be the key there.