How did scientists establish the reactions that occur in metabolism?
In addition to biochemical methods, a lot of this was figured out using radioactive tracers. Cells would be fed a radioactive substrate and as they convert the substrate to other molecules through metabolic reactions, the radioactivity can be detected in the products and intermediates.

Calvin's 1940s-50s experiments to establish carbon flow during photosynthesis (i.e. the Calvin-Benson Cycle) using $^{14}CO_2$ remain some of the most elegant work along these lines. You can actually read some of his reports here, though maybe someone else has a good link to a less technical overview.

Similar methods are still used today, though stable isotopes instead of radioactive ones are more commonly used.