Algebraic Geometry is a huge area of mathematics which went through several phases: Hilberts fundamental paper from 1890, sheaves and cohomology introduced by Serre in the 1950s, Grothendiecks theory of schemes in the 1960s and so on. This book covers the basic material known before Serres introduction of sheaves to the subject with an emphasis on computational methods. In particular, we will use Gröbner basis systematically. The highlights are the Nullstellensatz, Gröbner basis, Hilberts syzygy theorem and the Hilbert function, Bézout theorem,  semi-continuity of the fiber dimension, Bertinis theorem, Cremona resolution of plane curves and parametrization of rational curves. In the final chapter we discuss the proof of the Riemann-Roch theorem due to Brill and Noether, and give its basic applications.The algorithm to compute the Riemann-Roch space of a divisor on a curve, which has a plane model with only ordinary singularities, use adjoint systems. The proof of the completeness of adjoint systems becomes much more transparent if one use cohomology of coherent sheaves. Instead of giving the original proof of Max Noether, we explain in an appendix how this easily follows from standard facts on cohomology of coherent sheaves.  The book aims at undergraduate students. It could be a course book for a first Algebraic Geometry lecture, and hopefully motivates further studies.