Santos also expressed a desire to smooth over strained relations with neighbouring countries.
"One of my essential purposes as president will be to reconstruct our relations with Venezuela and Ecuador and to restore confidence and to make diplomacy and prudent actions a priority," he said.
Hugo Chavez, the Venezuelan president, responded to the Colombian president's offer, saying on national television that he too wanted to "turn over the page" and re-start good relations with Colombia's new government.
Chavez cut ties with Colombiaon July 22 after the government of Alvaro Uribe, Santos' predecessor, released evidence that it said showed Venezuela gives haven to Colombian rebels.
Although invited, Chavez decided not to attend the inauguration on Saturday, dispatching his foreign minister Nicolas Maduro instead.
Santos' inauguration ceremony in Congress was attended by about 5,000 guests, including many foreign heads of state.
He begins his term with a strong mandate after winning 69 per cent of the vote in the presidential runoff vote in June. He gained popularity during his 2006-2009 stint as defence minister, leading successful operations against Farc fighters.
Santos had travelled to northern Colombia early on Saturday to participate in a leadership ritual with representatives of four indigenous communities.
"Earth and water produce the food and now I understand what I have received: water, earth, food, the people and the government," he told the group of tribal leaders.
"That is going to be my inspiration during my term [as president]. I receive these elements with a great commitment and I will take good care of them."
Al Jazeera's Teresa Bo, reporting from Bogota, said this was the first time a Colombian president attended such a ceremony.
"Let's not forget that Colombia's indigenous communities have been hit by war," she said.
"They have been caught in the crossfire in the fights between paramilitary groups, left-wing rebel groups and even the army.
"So there's a very clear message in this trip. He [Santos] has spoken about a government of unity and he is obviously saying that he wants all Colombians to be part of it."
On the domestic front, Santos will have to deal with an unemployment rate of 12 per cent and 46 per cent poverty.
He has vowed to create 2.5 million new jobs during his four-year term in office.
He has also pledged to continue Uribe's crackdown on Farc and maintain his
pro-business approach, which has seen foreign investment grow five-fold since 2002 as Colombia's conflict waned.
Uribe leaves office with an 80 per cent approval rating.
But his second term was marred by scandals over abuses by troops, illegal wiretapping of his critics, and probes into legislative allies over collaboration with paramilitary gangs.