We present our current knowledge of the set L of geodesics in the Cayley graph of the first Grigorchuk group G. In particular, we present an indexed language that describes all the geodesics coming from geodesics in Schreier graphs. This language, as well as the full language of geodesics L, is closed under the endomorphism used in the Lysionok presentation of G. Further, we show that half of a Lysionok word is always a geodesic representing a dead end. We present some conjectures on flat tetrahedra and dead ends in the Cayley graph. Some of the difficulties in dealing with the set of geodesics L in G is apparent from the fact that L does not contain any infinite context-free subset.