How to unlock the herbalist quest

You should not need medicine before you get the herbalist. What happened? You unlock the herbalist with your first sickness on day 5

Charon is correct, the Herbalist quest will automatically unlock for you after a few days of play, and your wolves should be immune to illness until then.

If you're referring to Healing Salves (I see your lead wolf is low on HP)- some Explore encounters will drop them, and you can also buy or exchange them with other players in the Trading Center. I'll send you a couple to get you started!