Astronaut
The first civilian in space was Valentina Tereshkova aboard Vostok 6 (she also became the first woman in space on that mission).
Tereshkova was only honorarily inducted into the USSR's Air Force, which did not accept female pilots at that time. A month later, Joseph Albert Walker became the first American civilian in space when his X-15 Flight 90 crossed the line, qualifying him by the international definition of spaceflight. Walker had joined the US Army Air Force but was not a member during his flight. 
The first people in space who had never been a member of any country's armed forces were both Konstantin Feoktistov and Boris Yegorov aboard Voskhod 1.
Wally Funk
Funk always wanted to be an astronaut. She volunteered to be a part of the Women in Space Program in 1961, becoming one of the Mercury 13. Like the other participants in the program, Funk was put through rigorous physical and mental testing. In one test, volunteers were placed in sensory deprivation tanks. Funk was in the tank, without hallucinating, for 10 hours and 35 minutes, a record. She passed her tests and was qualified to go into space. She scored higher than John Glenn did in his testing and was the third best in the Mercury 13 program. Despite this, the program was canceled before the women were to undergo their last test.