Technically the first vaccine wasn't a vaccine, it was called virulation. It basically consisted of taking a bit of pus from a small-pox infected person and putting it in an open wound. It would give you a lesser form of the disease that would hopefully still allow you to survive (survival rate was 98-99%).
Until the ability to culture cells (which came later) was practiced, they practiced a technique called attenuation which involved taking a live disease and passaging it through animals (in the case of rabies, the strain was taken from cow's brain and passaged through rabbits) until the strain became attenuated, or non infected. They then took this attenuated, or avirulent strain, and used it as the vaccination so the immune system could build a defense for it.