This is a first graduate course in algebraic geometry. It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion of some of the technical foundations, but with no pretence at an exhaustive treatment. I hope that graduate students can use some of these chapters as a reader through the subject, maybe in parallel with a conventional textbook. The early chapters introduce topics that are useful throughout projective and algebraic geometry, make little demands, and lead to fun calculations. The intermediate chapters introduce elements of the technical language gradually, whereas the later chapters get into the substance of the classification of surfaces.