In this paper a study of the $d$-connection transformations on the total space $E$ of a vector bundle $\xi=(E,\pi,M)$ is given. Is 2. the $\tau$-and $\Omega(\rho)$-systems of tensor equations on $E$ are obtained, general solutions and some particular solutions are obtained, using the method given in [3]. Starting from these equations in 3. a general study of the connection transformations $\tau\colon D\to\bar D$ on $E$ is developed. Furthermore, special formulas of the classical projective transformations, which preserve the class of linear $d$-connections on $E$ are given. In 4. the transformations which preserve the torsion are studied. The study will be continued in the second paper (II) in which the transformations of linear $d$-connections with the invariants of Schouten and Weyl type will be studied.