Learn Spanish (Latin America): El doctor me da un medicamento.

Me is an indirect object pronoun, which tells us who receives the action (in this case, 'to me'). In Spanish, these pronouns are placed immediately before a conjugated verb. You will see this exact same pattern with verbs like gustar (as in me gusta) and doler (as in me duele).

Is dar (to give) considered a 'verb like gustar'?

No, dar is a normal action verb. However, it uses the exact same indirect object pronouns (me, te, le, nos, les) that verbs like gustar use. While me gusta means 'it is pleasing to me', me da means 'he/she gives to me'.

Could I use medicina instead of medicamento here?

Yes, absolutely! Un medicamento specifically means a medication or pharmaceutical drug. Una medicina can also mean a medication, but it can additionally refer to the science or field of medicine.

Does el doctor change if the doctor is a woman?

Yes! If the doctor is a woman, you would change both the article and the noun to say la doctora.