Folk singer Christy Moore has sang the final song - the Time has Come - at the graveside of Martin McGuinness' funeral.
Earlier, thousands of mourners heard ex-US President Bill Clinton say the people of Northern Ireland must finish the work he started.
Mourners included DUP leader Arlene Foster, who received applause as she took her seat in the church inside St Columba's Long Tower church.
After the service, thousands followed the coffin of Mr McGuinness onward to the republican plot of the City Cemetery.
Elected members of Sinn Fein formed a guard of honour as he was carried on his final journey.
Mr Adams said: "Here at the graveside of this good man let me appeal to our unionist neighbours.
"Let us learn to like each other, to be friends, to celebrate and enjoy our differences and to do so on the basis of common sense, respect and tolerance for each other and everyone else, as equals.
"Let me appeal also to nationalists and republicans, do nothing to disrespect our unionist neighbours or anyone else.
"Stand against bigotry. Against sectarianism. But respect our unionist neighbours.
"Reach out to them. Lead, as Martin led, by example."
Mr Adams challenged those who described Mr McGuinness as a "terrorist".
"Martin cannot answer them back, so let me answer for him," he said.
"Martin McGuinness was not a terrorist. Martin McGuinness was a freedom fighter.
"He was also a political prisoner, a negotiator, a peacemaker, a healer."
The Sinn Fein figurehead said it was not the case his long-time friend had a "road to Damascus conversion" and joined the political establishment.
He said throughout all he did he was guided by his republican principles.
"There was not a bad Martin McGuinness or a good Martin McGuinness," said Mr Adams.
"There was simply a man, like every other decent man or woman, doing his best.
"Martin believed in freedom and equality.
"He resisted by armed actions those who withheld these rights, and then he helped shape conditions in which it was possible to advocate for these entitlements by unarmed strategies.
"Throughout it all Martin remained committed to the same ideals that led to his becoming a republican activist in the first instance, the pursuit of Irish unification, freedom, equality and respect for all."
He added: "Our political objectives, and our republican principles and ideals did not change.
"On the contrary these guided us through every twist and turn of the peace process.
"Thanks to Martin we now live in a very different Ireland, which has been changed utterly.
"We live in a society in transition. The future now can be decided by us. It should never be decided for us.
"Without Martin there could not have been the type of peace process we've had.
"Much of the change we now take for granted, could not have been achieved.
"In my view the key is in never giving up. That was Martin's mantra also."
Irish President Michael D Higgins and his predecessor, Mary McAleese, attended the funeral, as did Taoiseach Enda Kenny and former Irish premiers Bertie Ahern and Brian Cowen.
Northern Ireland's police chief, George Hamilton, was also there.
Ahead of the service, Mr McGuinness's beloved Bogside neighbourhood came to a standstill as his remains - the coffin draped in the Irish Tricolour - were walked to St Columba's Church, led by a lone piper.
Bishop Donal McKeown, Bishop of Derry, opened requiem mass by welcoming dignitaries, public figures and politicians from Ireland, Britain and the US.
He then turned to the McGuinness family and said: "For you, this is not the funeral of a public figure.
"This is a funeral of a husband, father and a grandfather and our first thoughts are inevitably with you."
Bishop McKeown noted the applause inside St Columba's when some politicians had arrived ahead of Mr McGuinness's remains, including for Mrs Foster.
And he thanked those who had been involved in securing the Good Friday Agreement who travelled for the mass, including from within Northern Ireland and the Republic and Britain and the US.
"It's a tribute to those who didn't just talk the talk but walked the walk of implementing the Good Friday Agreement that all three of those strands are so well represented here," he told mourners.
Inside, chief celebrant Father Michael Canny began his homily by asking mourners to join him in sharing thoughts and prayers with the people of London following Wednesday's terror attack.
He recalled the many tributes made to the Sinn Fein politician since his death and said it has been acknowledged that Mr McGuinness spent year after year moving the community towards peace.
"There are people in this church today whose presence would have been unthinkable only a generation ago," Fr Canny told mourners.
The congregation heard that the presence of Mr McGuinness's political rivals and opponents at the mass is "the most eloquent testimony" to his memory.
"When you seek his monument, look around you. You, by your presence, are his monument," Fr Canny said.
Mourners were told Mr McGuinness was the IRA commander who became a mainstay of the peace process.
Fr Canny revealed having many conversations with Mr McGuinness in which the Sinn Fein veteran said he knew only too well how many people struggled with his IRA past.
"Republicans were not blameless, and many people right across the community find it difficult to forgive and impossible to forget," he said.
"By any standards, Martin McGuinness was a remarkable man and his life was a remarkable journey. The values he had, the principles he championed are still very much alive.
"On that journey many years ago, Martin realised that the time for peace had come and he pursued the peace process with relentless energy for the rest of his days, until illness finally struck him down.
"In the course of that journey he encountered many obstacles but he remained resolute. In conversation, he often repeated that there was no other way, we had to continually work for the building of peace and a better future for all.
"Despite many setbacks, he never became disheartened."
Among others to pay tribute during the mass were two Protestant ministers - Reverend Harold Good, former president of the Methodist Church in Ireland and a member of the panel which oversaw IRA decommissioning, and Rev David Latimer of the First Derry Presbyterian Church.
Rev Latimer described Mr McGuinness as a friend and recalled praying with him.
"Martin had a good heart but I would go a bit further to say, he had a big heart that enabled him to reach out in quite unexpected ways to both to friends and to foes alike," he said.
"Martin has bequeathed to us a better place to live.
"It was his commitment to create a new order of co-operation where we will be able to live in relationship and not out of relationship and get to know each other better.
"In memory of the man whose friendship I will always treasure we must together, all of us, pledge to keep on doing what he was doing and to persevere in the pursuit of peace.
Rev Latimer added a message of thanks in Irish: "Go raibh mile maith agat, Martin."