Mr Davis and Mr Barnier gave a joint press conference after day one of the Brexit negotiations in Brussels. The initial focus will be on expat rights, a financial settlement and "other separation issues".
Michel Barnier said he was "not in the frame of mind to make concessions or ask for concessions", which the Brexit Secretary David Davis said talks got off to a "promising start".
The UK appears to have conceded to the EU's preferred order for the talks which will mean trade negotiations do not begin immediately, with Barnier saying "basically, we are implementing the decision taken by the United Kingdom to leave the European Union, and unravel 43 years of patiently-built relations.
The initial focus will be on expat rights, a financial settlement and "other separation issues".
Discussions aimed at preserving the Good Friday Agreement and common travel area in Ireland will also begin, although Mr Davis suggested these issues may not be settled until the end of the process, when the UK's trade relationship with the EU is settled.
The UK had wanted talks on its future relationship with the EU to be considered from the outset, but Mr Barnier said this would only happen once the European Council decided "sufficient progress has been made" on the other issues.
Mr Davis - who had predicted this would be the "row of the summer" - denied suggestions the agreed timetable showed Britain's "weakness" and insisted it was "completely consistent" with the government's aim of parallel trade and exit talks. "It's not when it starts it's how it finishes that matters," he said.
London School of Economics professor Silvana Tenreyro has been appointed as the newest external member of the Bank of England's monetary policy committee (MPC). Tenreyro, who teaches macroeconomics and international economics, will replace noted hawk Kristin Forbes on the rate-setting MPC
Mark Carney has said that Tenreyro will begin her three-year term on 7 July. In an interview in January, Silvana said she was pessimistic about Brexit, and said the MPC's decisions will be based on the outcome of negotiations.
Households in Britain have become more worried about the outlook for their finances in the 12 months ahead as rising inflation puts a squeeze on their spending power, a survey showed on Monday.
IHS Markit said its index measuring how households feel about their personal finances fell to 45.8 in June from 47.1 in May, the most pessimistic in three months and one of the lowest readings since the end of 2013.
The IHS overall Household Finance Index, measuring how people feel about their current situation, rose to 43.8 from 42.6 but remained below the 50.0 no-change level.