I love Classic fiction and, like you, have been reading it since a young age. I always have trouble narrowing down my book love to just one or two 'favourites', or 'bests' (or ones that moved/inspired/changed me) - it totally depends on my mood and the day.
I am a huge fan of George Eliot so will always have her at, or very near, the top of my most loved Classics list. Alexandre Dumas is another long-time favourite, and I am a total sucker for Classic Russian lit. - especially in winter.
But, in an attempt to answer your question, I am going to say that my experience with reading A Tale of Two Cities, by Charles Dickens will always be a treasured memory. I read the book in high school, and was very lucky to have a great and inspiring teacher taking us through the novel. And what a story!! My teacher's love for Classic literature was so evident - it made me less... concerned about my own bookishness. Being a book nerd, at that time, was not quite the awesome thing it is today. I have read a lot of Dickens since that time, but A Tale of Two Cities will probably always be my favourite of his works.