How does a Buddhist care for others' suffering?
When a person is sick, due to knowing they are sick, they act to visit a doctor. To be a doctor, a person must study medicine at university for at least 5 years, before they can help sick people.

Similarly, a person who is suffering must, via their own initiative, seek out a solution. Similarly, to be a Buddhist that can help others end suffering, a person must train for enlightenment. 

An enlightened Buddhist makes themselves available to help others. However, others must learn to help themselves by following the advice of Buddhism.