Can anyone explain why most acids and salts are soluble in distilled water and bases generally are not? I find this very curious... Thanks!

bases are too, but it depends on the base.
it's a matter of molecular dissociability

that is, how readily the cation dissociates from the hydroxyanion

Thanks for the reply Benzyme. I understand that solubility and dissociation constants are a measurement of how well things do dissolve, but my curiosity is more oriented towards the physical reason why salt and acid molecules tend to dissolve readily in polar solvents and bases tend to dissolve in NPs... maybe I'm just misled by the particular acids and bases I'm familiar with, which certainly don't represent the total spectrum of acids and bases. Is there a simple intuitive reason for this?

bases do not dissolve in nonpolars, they dissolve in polars.
a strong base is one that completely dissociates in water.
I know this seems counterintuitive, because NaOH is difficult to dissolve in water yet is considered a strong base.. CsOH and LiOH are other strong bases that dissolve easily in water.

oh sorry my wording is confusing... I mean alkaloid bases vs the acids used to salt them. thanks!

Also whatever actually does dissociate, COMPLETELY dissociates in water therefore it's a strong base, and it works exactly the same for acids.

goes without saying

pH and pOH, determined from difference of pKa and pKb,respectively, from the ratio of concentration of product over conc. of reactants in sol'n.
classic henderson-hasselbalch eq.
fundamental analytical chem

Why is DMT freebase soluble only in a nonpolar solvent while DMT salt is soluble only in a polar solvent?

What is happening at the particulate/molecular/atomic level when NaOH is added to an aqueous solution of say DMT tannate?

the base dissociates in the solution, and the hydroxide ions deprotonate the salt, forming sodium tannate salts (which stay in the aqueous phase), and dmt free base. the free base, now uncoupled from the otherwise water-soluble salt complex, carries a net neutral charge..
the overall molecule is nonpolar because of the conjugated double-cyclic structure of indole; because of this, it will migrate towards a nonpolar solvent.

There are numerous acids and salts that do not dissolve in water. I think the inclination to propose dichotomies like "acids and salts dissolve in water, bases do not..." etc are biased because stem from personal experiences e.g. working with extractions.

Acids and salts that do not dissolve in water are things like fatty acids and their salts. Few very common examples are octanoic acid and calcium octanoate and alpha acids (the bittering agents in hops).