To find the union of sets A and B, and the intersection of sets B and C, using the given sets.

The union of two sets contains all unique elements that are in either set.

$$A \cup B = \{5, 10, 12, 15, 19\} \cup \{3, 10, 15\}$$ $$A \cup B = \{3, 5, 10, 12, 15, 19\}$$

The set $A \cup B$ is $\boxed{\{3, 5, 10, 12, 15, 19\}}$.

The intersection of two sets contains only the elements that are common to both sets.

$$B \cap C = \{3, 10, 15\} \cap \{7, 10\}$$ $$B \cap C = \{10\}$$

The set $B \cap C$ is $\boxed{\{10\}}$.