In theory, you can grow bonsai in just about any medium if you understand how to water it and how the medium interacts with your tree's roots, with the water you use and with the fertilizer you use. The effects of the media change depending on species and age of the tree. So there are no hard and fast rules, and there are at least as many good media as there are bonsai growers and species of trees to custom blend individual mixes for. A thorough discussion would require hours of typing. Please go through the various threads here on BNut to read various formulations. One by Smoke (Al Kepler) is particularly good. I don't have the link.
In practice some media are easier to use than others. One of the most commonly used components is pumice, a volcanic rock that has to be shipped a long way to get it to Long Island, NY. Only for special purposes is it 100% of the mix. All of the most successful professional mixes use some fraction of pumice. The Japanese use pumice for bonsai, especially for pines and conifers.
A standard mix is roughly 2 parts pumice, one part composted douglas fir bark (seedling grade orchid bark), 1/10th part horticultural charcoal.
There are 1000 variations on this simple 3 part blend.
Akadama is a clay imported from Japan that the Japanese use, often blended with pumice to make it less water retentive. Kanuma is another imported Japanese volcanic clay that is used for bonsai. These are both clays derived from volcanic origins and are chemically distinct from non-volcanic sourced clays. In other words Turface is not a substitute for Akadama. No commercially available clay product in north america is a good substitute for the 2 Japanese sourced clays. But if you don't want to mail order in Japanese media, there are media that grow good Bonsai, without Akadama or Kanuma. Pumice is the best main ingredient of the mixes.
Pumice - light volcanic rock, differs from lava in that there are many more pores in pumice, making much lighter, less dense. If you can not get pumice one possible substitute is perlite, also known as sponge rock. Difficult to source perlite with large enough particle size. Order from orchid supply house. They carry a more coarse perlite.
Taxodium - Bald Cypress, - loves a mix heavy in organics, usually most people grow them in 50% composted fir bark. Rest is pumice. Some add modern designed potting mixes that are peat, or coir based products like Promix or Farfaad BX. Should be available at local landscape nurseries. Be certain to buy one or the other brand. ''Potting Soil'' can be any awful mix, but Promix and or Farfaad do make professional grade products. Many advocate growing bald cypress in a tray of standing water. A commercial deciduous bonsai mix would work, especially if you added extra fir bark, or some potting mix (Promix or Farfaad) to bring organic phase up to about 50%
Metasequoia - similar in its love of organics in its mix as bald cypress, but does not want to be as wet. Here the original proposed general mix of 2 parts pumice one part composted bark is ideal. A commercial deciduous bonsai mix would work fine (even though the metasequoia is classed as a conifer)
Beech - Fagus - this is a deciduous tree and deciduous bonsai mix would work well, basically the ''standard mix''. Fagus does prefer somewhat acidic soil similar to what azalea like, so if you are familiar with azalea you could try using an azalea mix. Commercial deciduous bonsai mix is just fine. They are tolerant enough of pH that usually you don't have to worry much about keeping it slightly acidic.
Cupressus - the true cypresses. These are conifers and like a somewhat dryer or more aerated mix. Here 3/4 pumice, 1/4 composted fir bark would be better. You can use any commercial bonsai mix designed for pines.
So
Does this give you a starting point?
There are many ways to have success, so don't get frustrated when trying to read through all the ''stuff'' written about soils. Ignore the ''Soil Wars'', many are passionate about what they feel works best.
Read threads in this sub forum
https://www.bonsainut.com/forums/soil-and-fertilizer-incl-water-and-repotting.63/
