How did Han China (206 B.C.E.--202 C.E.) react to the introduction of Buddhism?

The Buddhist faith did indeed enter China during the Han
Dynasty.  Scholars are not completely sure when it first got to China, but it is
believed to have been some time in the first century of the Common
Era.

The Han Dynasty had no problems with Buddhism.  It did
not try to resist Buddhism at all and the faith spread fairly rapidly in China. 
Scholars believe that Buddhism took hold more strongly and quickly among the elites of
the northern parts of China and then spread to the South and to other parts of
society.

The Chinese are said to have adopted Buddhism
quite easily because the Buddhist faith was compatible with (and willing to adapt to)
indigenous Chinese religions.  Buddhism typically became somewhat synthesized with both
Taoism and Confucianism and thereby managed to make its way quite easily into Chinese
culture.