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Brexit: Theresa May says talks won't start in 2016
Earlier, a military guard of honour greeted Mrs May, who succeeded David Cameron a week ago.
At a joint press conference, Mrs May said the UK was in no rush to trigger the two-year process of leaving the EU - telling reporters that although "this would not please everyone" it was right to hold off until the UK's "objectives were clear".
The process of preparing the UK for Brexit would require "serious and detailed work" but, irrespective of this, she said the UK was determined to maintain strong trading, economic and security links with Germany, which she described as "a vital partner and special friend".
"Of course, the nature of our relationship is going to change as the UK leaves the EU, but we both want to maintain the closest possible economic relationship between our countries and I believe that is what German and British businesses want too," she said.
"So it's good that we start from such a strong foundation and a position where both our countries believe in liberal markets and free trade and these should be the principles that guide us in the discussions ahead."
Asked how they had got on at their first meeting, Mrs May said they were two women who want to "get on with the job and deliver the best possible results for the people of the UK and Germany".
Mrs Merkel said she did not expect there to be any formal negotiations at this stage and it was "understandable" the UK needed a period of time to prepare.
But she said there was a need for a "certain timeline" with regard to Britain's exit and hoped the UK would begin to "define its principles" with regard to the process of activating Article 50 of the Lisbon Treaty, the official, legal mechanism for leaving.
"We have to listen to what Britain wants and find what the right answer is," she said. "Britain does not want an impasse, Germany does not want an impasse and the EU does not want an impasse".
The German chancellor said the two countries had "similar convictions and values" and she was determined to proceed in the "spirit of unity and friendship" that characterised Anglo-German relations.
BBC political editor Laura Kuenssberg said it was hard to believe that it was the first time that the two women had properly met as they seemed immediately at ease in each other's company and determined to build a personal relationship based on mutual trust.
But she said it was clear that Germany would hold the "balance of power" in the negotiations to come and although Mrs Merkel did want to make the UK's departure work, the UK's "leverage had really faded" following the decision to leave.
A Downing Street source said the government was "encouraged" by the tone of the talks and the subsequent working dinner in the German chancellery.
"There was a free flowing conversation in the bilateral talks and the dinner," the source said. "The two established a personal connection and relationship and the 'communication channels are now open'.
It has emerged that the UK prime minister presented two books on the British outdoors to her German counterpart as a birthday present, reflecting their shared interest in hiking.
The gifts for Mrs Merkel, who turned 62 on Sunday, were a new edition of Coast To Coast With Wainwright - a pictorial guide to illustrate Alfred Wainwright's walking route between northern England's west and east coasts - and a copy of Great Mountain Days In Snowdonia, which includes a guide to walks in the National Park.
Mrs Merkel often holidays in the South Tyrol region of northern Italy, while Mrs May is reported to enjoy hiking holidays in the Alps.
Mrs May is due to have talks with France's Francois Hollande on Thursday. Ahead of the visits - Mrs May's first overseas trips as prime minister - Downing Street announced the UK was to relinquish its upcoming six-month presidency of the Council of the EU.
The UK had been scheduled to take up the presidency of the Council of the EU - which rotates on a six-monthly basis between the 28 EU countries, giving each the opportunity to shape the agenda - in the second half of 2017.
But Mrs May has decided that Britain should skip its turn in the light of the Brexit vote in June's referendum.
Mrs May told European Council President Donald Tusk - in her first conversation with him as PM - it was "the right thing to do given we will be very busy with negotiations to leave the EU", a Downing Street spokesman said.
While the German and French leaders have said the UK's vote to leave must be respected, both are facing re-election next year and under domestic political pressure to drive a hard bargain.
They have suggested no special exceptions can be made for Britain in terms of continued access to the EU's single market if, as Mrs May has insisted, the UK seeks controls on freedom of movement rules.
The first time that Mrs May will face all 27 other EU leaders at the same time will be at October's European Council meeting.