In these papers we deal with geometry of group and group actions on graphs and metric spaces. Our objective is to observe infinite, finitely generated groups, because there are no strong results about structure of those groups in standard group theory, except the main result about finitely generated Abelian groups. To begin with, we deal with some basic results in standard group theory, because we want to improve such basic results using methods of geometric group theory. In general, we observe group actions on connected graphs, especially trees. We introduce notion of Cayley graph of group and deal with some important results about group structure. We prove some teorems about caracterization of free groups, by looking on group action on trees and Nielsen-Schreier’s teorem about subgroup of free group. Furthermore, we define what is geometry of group. We give every group and its set of generators some metric space, with metric which strongly depends on group structure. This metric, we call word metric on some group, regarding given set of generators. Only problem is that this metric strongly depends on set of generators. Therefore, we introduce a criterium in order to differentiate metric spaces up to quasy-isometry, so that every finite set of generators will induce isomorphic metric spaces in given category. Furthermore, we observe isometric group action on quasi-geodesic spaces. After that, we prove famous Švarc-Milnor lemma and deal with rigid or geometric group properties. Such properties are some properties of groups wich are invariant regarding quasi-isometry. We especially observe growth of groups, as important geometric property of groups and we deal with famous Bass, Wolf, Milnor and Gromov theorems and their consequences. In the end, we observe hyperbolic and quasi-hyperbolic spaces and hyperbolic groups in order to give some application of those notions on so-called Word problem.