  We study the Chow ring of the moduli stack $\mathfrak{M}_{g,n}$ of prestable
curves and define the notion of tautological classes on this stack. We extend
formulas for intersection products and functoriality of tautological classes
under natural morphisms from the case of the tautological ring of the moduli
space $\bar{\mathcal{M}}_{g,n}$ of stable curves. This paper provides
foundations for the second part of the paper.
  In the appendix (joint with J. Skowera), we develop the theory of a proper,
but not necessary projective, pushforward of algebraic cycles. The proper
pushforward is necessary for the construction of the tautological rings of
$\mathfrak{M}_{g,n}$ and is important in its own right. We also develop
operational Chow groups for algebraic stacks.
