High accuracy of major biological processes relies on the ability of the participating enzymatic molecules to preferentially select the correct substrate from a pool of chemically similar substrates by activating the so-called proofreading mechanisms. While the importance of such mechanisms is widely accepted, it is still unclear how evolution has optimized the biological systems with respect to certain characteristic properties. Here, using a discrete-state stochastic framework with a first-passage analysis, we theoretically investigate trade-offs between four characteristic properties of enzymatic systems, namely, error, speed, noise, and energy dissipation. Specifically, two fundamental biological processes are examined, i.e., DNA replication in the T7 bacteriophage and tRNA selection during protein translation in Escherichia coli. Notably, all of the characteristic properties cannot be completely optimized at the same time due to trade-offs between them. To understand the relative importance of the computed quantities to the enzymatic functionality, we introduce a new quantitative metric to rank the properties. The results demonstrate that the reaction speed is the principal characteristic property that evolution optimizes in both enzymatic systems and that the energy dissipation comes in second. In addition, the error and the noise are always ranked third and fourth, respectively, regardless of the system considered. Physicochemical arguments to explain these observations are presented.