How does a layman such as myself 'test solubility' of given materials?
you measure it empirically
figuring out solubility figures for single compounds won't necessarily translate to mixtures, e.g. just because the solubility of, say, sugar or salt in water is a certain amount, doesn't mean you can dissolve that amount of e.g. sugar in a concentrated brine solution.
Since its so cheap it doesnt have to be that accurate does it so just keep extracting till the final extractions are clearer? Is that a good enough way?
that would be a good method. unless the compounds you're interested in are both colorless & have a high affinity for the plant material.
What are the general chemistry principles involved so I can go look them up myself.
well, you don't really need a knowledge of chemistry, more like handling it as a word problem. The general rule of thumb is, for substances that dissolve easily, 3 extractions is enough.
If every extraction pulls 80% of the juice out then
1st extraction will remove 80% = 80% total
2nd extraction will remove 80% of the 20% remaining (16% ) = 96% total
3rd extraction will remove 80% of the 4% remaining (3.2% ) = 99.2% total
4th extraction will remove 80% of the 0.8% remaining (0.64% ) = 99.84% total