Coote was born in Melbourne, and studied at Fintona Girls' School and the University of Melbourne. Before being elected to parliament, she worked in a number of business and executive positions, which included stints as Executive Director of the State Library Foundation (1994-1996) and Director of Development at Geelong Grammar School (1996-1998). In the early 1990s, she also worked as a staff officer to Member of the Legislative Assembly Michael John, and later as a ministerial adviser.
Coote succeeded in winning Liberal pre-selection for the theoretically safe Liberal-held seat of Monash Province in the leadup to the 1999 election, and was ultimately elected. Jeff Kennett's Liberal government, however, had suffered a surprise defeat, so Coote found herself in opposition. Despite this, due to the way the Legislative Council is elected, the Liberal Party still had control of the upper house, giving them significant power to reject legislation introduced by the new Labor government. In her first term in office, Coote was a member of the Legislative Council Printing Committee, and in 2000, also gained a seat on the Economic Development Committee.
During her first term, Coote also became notable for her support of the lesbian, gay, bisexual and transgender communities. While the Liberal Party, as a conservative party, has traditionally opposed any liberalisation in this area, Coote became one of a small group of Liberal Legislative Councillors - along with Andrew Olexander (since expelled from the parliamentary Liberal Party following a drink-driving incident), and before they lost their seats at the 2002 election, Peter Katsambanis and Leonie Burke - that lobbied for gay-friendly changes in party policy. Along with these three, Coote regularly met with representatives from the LGBT communities in Melbourne. The Victorian Gay and Lesbian Rights Lobby specifically singled out Coote for praise, describing her as "very proactive on gay issues". The influence of Coote and her allies was most clearly illustrated when the Liberal Party backed down on its prior opposition to the 2001 Statute Law Amendment (Relationships) Bill, which Coote spoke in support of. The passage of the bill resulted in major changes, requiring that lesbian and gay relationships be treated equally in fields such as property, superannuation, and medication decision-making.
The 2002 election was devastating for the Liberal Party, resulting in the loss of their control of the Legislative Council, and with Katsambanis and Burke being among those losing their seats. However, this opened up a number of new shadow ministerial positions, and Coote was one of those promoted. She became Shadow Minister for Tourism, Deputy Leader of the Opposition in the Legislative Council, Manager of the Opposition Business in the Council and Shadow Minister for Ageing and Carers. She also received the additional portfolio of Victorian Communities in January 2004.
Coote was re-elected in 2006 as a Member for Southern Metropolitan Region
In 2006, Andrea was appointed Shadow Minister for the Portfolios of Community Services, Children and Aboriginal Affairs. She is also Manager of Opposition Business in the Legislative Council, and is a member of the Legislation Committee of the Legislative Council. In 2007 she was reelected as Deputy Leader of the Legislative Council for the Opposition.