Stargirl is a character who, since her inception, has been all over the DCU. Here is a list of easy to read, yet essential, material for the character.
Stars and S.T.R.I.P.E. This is the series that marks the debut of both Stargirl and her creator Geoff Johns in DC. It tells the story of a new step father & daughter crime fighting team, as they dive into the world of super heroes. It's a fairly good series and a great intro to the character.
JSA #81: My Heroes If you want one issue to sum up and showcase Stargirl as a character, then this comic does a pretty good job. It's a nice one and done story about Stargirl that anyone can pick up and read. It's also written by her creator Geoff Johns.
Justice League of America: Survivors of Evil This story takes place in the middle of the Forever Evil event, but it's a very self contained story that has little impact on the event itself, so it is still relatively easy to read. The story focuses on Stargirl and Martian Manhunter, it's very introspective, and establishes Stargirl's character and background for the New 52.
If after reading these you want to continue keeping up with the character, then at the time this is currently being written, Courtney is a part of Jeff Lemire's Justice League United series. So far it's a fairly well received series that's ripe for jumping on and reading.