Although the Brexit secretary said he had been encouraged by the constructive approach both sides had taken, the joint press conference hinted at the difficulties ahead.
The EU is also seeking British commitments to meet all past EU spending pledges when it leaves the bloc, a bill European officials say could amount to upward of EUR60 billion ($67.2 billion).
They said discussions would be split into three stages: citizen rights, the single financial settlement and other separation issues.
Though less visibly upbeat than veteran Brexit campaigner Davis, Barnier insisted the two sides would work together for a "fair deal" that would not "punish" Britain.
Britain now appears to have given in on the EU's insistence that the negotiations first focus on three key divorce issues, before moving on to the future EU-UK relationship and a possible trade deal.
"We want both sides to emerge strong and prosperous, capable of projecting our shared European values, leading in the world, and demonstrating our resolve to protect the security of our citizens".
Barnier said he was hoping to have a clearer timetable by the end of the day. "It is not when it starts, it is how it finishes", he said.
"They should be agreed alongside each other, this is completely consistent with the Council's guidelines which state nothing is agreed until everything is agreed".
Officials are hopeful that there can be swift progress on some areas of the border problem but Mr Davis acknowledged the issue as a whole "will take some considerable time". Led by Germany, Brussels had insisted from the beginning that Britain's decision past year to leave Europe clearly meant it was leaving Europe, with all the attendant privileges.
Asked if he had given any ground to Britain, Mr Barnier said: "I am not in a frame of mind to make concessions, or ask for concessions". "It's not about punishment, it's not about revenge". In mid-May, he claimed: "You can't decide one without the other".
Other issues on the agenda, at least in the near term, include establishing residential rights of more than 3 million European Union citizens now living in the United Kingdom - and the 1 million British expats living in European Union member states - once the formal exit is complete.
Finance minister Philip Hammond confirmed Sunday that it was still the plan to quit not only the EU but the customs union and the bloc's single market as well.
"So, we each have to assume our responsibility and the consequences of our decisions".
The EU has said that before it will start discussing a future trade deal with the United Kingdom, it wants to resolve the biggest uncertainties arising from Brexit, including the future rights of around 3 million EU citizens in the United Kingdom and the arrangements for avoiding a hard border between Northern Ireland and the Republic of Ireland.
For his part Mr Barnier presented Mr Davis with a hiking stick.