How did early chemists determine the major product of a solution?
In absence of currently available (or, in principle available) spectroscopies like NMR, IR, Raman, UV-Vis; and mass spectrometry our ancestors tried first to separate the products obtained by sequential recrystallisation.  Or by extraction under acidic, neutral, basic conditions (including back extraction).  

They then recorded other physical properties of the purified products, if accessible, which then were compared with reference data.  For example, melting point and crystal shape if the products were solid, otherwise boiling point, refractive indices of the products obtained.  Then, they knew a whole array of pretests to check for the presence / absence of elements in the products formed, like Beilstein test for halogens and Lasaigne test for nitrogen.  They would determine the molecular weight by means of cryoscopy or ebullioscopy.

From a chemical point of view, it was common to subject the products to combustion analysis to determine the molecular formula.  (One of the instruments for this, the  Kaliapparat is still present in the logo of The American Chemical Society.)  Till today, there are journals preferring this over a mass spectroscopic determination of the molecular weight.

They knew about reactions that would yield derivatives, for example from the reaction with picric acid (affording pircrates), or with toluenesulfonic acid (yielding amides); often crystalline with a sharp melting point, easing to compare with reference data.  If not, other reactions like the haloform reaction would sequentially degrade the products into fragments, and then these products obtained would be characterised by physical or chemical means.

And obviously, the balance was much more "sacred" than today, where often we "just" weight-in reactants, and weight-out products.