WASHINGTON—The Senate unveiled final details of a broad tax bill—and its 10-year price tag of $858 billion—and began debate Thursday night on the package, a significant step after two years of gridlock over how to treat expiring tax cuts enacted under former President George W. Bush. Without action, income taxes on nearly every American are due to rise on Jan. 1.

The Senate has moved quickly after Mr. Obama and Republican leaders unveiled the package Monday, with a key test vote now scheduled for Monday. The bill is expected...