A drill bit includes a bit body having a face on which two different types of cutting elements are disposed, the first type being cutting elements suitable for drilling at least one subterranean formation and the second type being cutting elements suitable for drilling through a casing bit disposed at an end of a casing or liner string and cementing equipment or other components, if such are disposed within the casing or liner string, as well as cement inside as well as exterior to the casing or liner string. The second type of cutting elements exhibits a relatively greater exposure than the first type of cutting elements, so as to engage the interior of the casing bit and, if present, cementing equipment components and cement to drill therethrough, after which the second type of cutting elements quickly wears upon engagement with the subterranean formation material exterior to the casing bit, and the first type of cutting elements continues to drill the subterranean formation. The first type of cutting elements may comprise superabrasive cutting elements and the second type of cutting elements may comprise abrasive or superabrasive cutting elements comprising a plurality of configurations.