What are some general tests that would determine an unknown compound?
As usual, it depends on what is available. 

Mass spectrometry, $^{1}\ce{H}$ and $^{13}\ce{C}$ NMR, IR, UV/VIS spectroscopy, would be obvious choices in organic chemistry nowadays. XRD of a suitable crystal would be nice to have! 

Back in the days, (undergrad) laboratories didn't have sophisticated analytical instruments and you were happy to find a melting point apparatus, a refractometer and an IR spectrometer.

You would have used IR spectroscopy to identify funcional groups, possibly supported your hypothesis by using staining reagents for TLC, like Dragendorff's reagent and then started to synthesize derivatives from which you knew that they show sharp, well-defined melting points. 

You would have converted  


amines to sulfonamides
carbohydrates to osazones  
ketones and aldehydes to 2,4-dinitrophenylhydrazones or semicarbazones


and compared the melting points of the derivatives with tabulated values. Those were the days&hellip;