The textbook covers some basic areas of graph theory: trees and forests; routes following the edges of a graph; routes covering the vertices of a graph; matching problems, factors; extremal values, extremal graphs. The approach differs from the classical one. Each chapter starts with exercises and problems. There follows a free exposition of the theory: analysis of the problems, remarks, particular solutions, general statements, solutions of problems, suggestions of new problems and generalizations, well motivated introduction of new concepts, new problems etc. This arrangement seems to be very natural and very stimulating, at least for those who want to enjoy mathematical thinking and to solve problems. Second volume (more advanced) is planned to appear.