The ceremonies are taking place quietly, a day before Obama and Biden take their public oaths in front of the Capitol before a crowd on the National Mall.
The Constitution calls for the president to be sworn in on January 20.
Biden took his official oath shortly after 8 a.m. at the Naval Observatory, his official residence. Supreme Court Justice Sonia Sotomayor administered the oath. With his hand on a family bible, Biden vowed to support and defend the Constitution.
Obama and Biden traveled to Arlington National Cemetery after Biden's swearing-in for a traditional wreath-laying ceremony.
The president's address Monday will lay out the values and vision for his second term and acknowledge the division in Washington but won't address policy, a source with knowledge of the speech told CNN on the condition of anonymity.
The president will lay out policy in his February 12 State of the Union address, the source said.
The nation's first African-American president also will become only the 17th U.S. leader to deliver a second inaugural address before joining the traditional parade up Pennsylvania Avenue to the White House.
For his second inauguration, less than half of the estimated 1.8 million onlookers who crammed the Mall in 2009 are expected -- organizers expect 800,000 people to attend Monday's public ceremony.