The easiest procedure would be to make tea from cinchona bark – enough quinine can be extracted this way to make a therapeutically useful dose for malaria treatment.
Making a reasonably potent antibacterial compound is much more difficult. The natural ß-lactams, such as penicillin, are not chemically very stable, so you would have a very hard time (as indeed Fleming and others did) in making enough to actually treat patients.
Your best bet might be to prepare an aminoglycoside such as streptomycin. They are chemically stable and are produced at high concentrations by an organism (Streptomyces griseus) that is easy to culture.
Easy for us, that is. The first practical methods for pure culture were developed by Robert Koch in the 1890’s, so if you were starting out 300 years ago you would have to formulate your own growth medium. Gelatin (extracted from pigs’ hooves) fortified with beef broth and perhaps milk might work. You could streak out samples of soil on these plates and look for S. griseus colonies. Identifying them would be hard, as this species has many ecovars. Perhaps an antibiotic streak test might work best
From Microbiology Laboratories
Once identified and isolated, a batch culture could be grown up in beef or milk broth. The culture could be clarified over charcoal (the antibiotic is excreted from cells, so you want to get rid of the cells and keep the supernatant).
To purify and concentrate it, the easiest method would be alkali precipitation from acetone.
According to Gorham and Doering (via Wikipedia) “Acetone was first produced by alchemists during the late Middle Ages via the dry distillation of metal acetates (e.g., lead acetate, which produced “spirit of Saturn”)”, so you would be OK there – just find some alchemists to make it for you. Vinegar might acidify the medium sufficiently to keep it in solution while you filter out the crud over charcoal. Then you would just need an alkali in order to precipitate out the streptomycin. A strong lye solution (sodium/potassium hydroxide) might be feasible. At that point you would pour off the acetone and let the precipitate air-dry.
It would probably take a year to actually make this work. You would have an excellent remedy for tuberculosis as well as plague. Women will adore you, men will want to be you. But priests and preachers will want to burn you.