A well-studied continuous model of graphs, introduced by Dearing and Francis [Transportation Science, 1974], considers each edge as a continuous unit-length interval of points. For $δ\geq 0$, we introduce the problem $δ$-Tour, where the objective is to find the shortest tour that comes within a distance of $δ$ of every point on every edge. It can be observed that 0-Tour is essentially equivalent to the Chinese Postman Problem, which is solvable in polynomial time. In contrast, 1/2-Tour is essentially equivalent to the Graphic Traveling Salesman Problem (TSP), which is NP-hard but admits a constant-factor approximation in polynomial time. We investigate $δ$-Tour for other values of $δ$, noting that the problem's behavior and the insights required to understand it differ significantly across various $δ$ regimes. We design polynomial-time approximation algorithms summarized as follows: (1) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour admits a constant-factor approximation. (2) For every fixed $δ\geq 3/2$, the problem admits an $O(\log{n})$-approximation. (3) If $δ$ is considered to be part of the input, then the problem admits an $O(\log^3{n})$-approximation. This is the first of two articles on the $δ$-Tour problem. In the second one we complement the approximation algorithms presented here with inapproximability results and related to parameterized complexity.