The impingement cooling for a modern gas turbine component, either a combustor liner or a high-pressure turbine blade, is often not as efficient as required due to strong cross-flow effect and coolant maldistribution. This paper reports a novel multi-stage impingement cooling scheme to effectively use the coolant and minimize the cross-flow effect. The design concept and general working mechanism are introduced in this Part I paper. The extra design flexibilities and optimization strategies are reported in Part II. Numerical simulations on conjugate heat transfer (CHT) were carried out to assess the flow structure and thermal performance between a typical single-stage cooling design and a three-stage cooling design at typical operating conditions. It has been observed that the novel multi-stage cooling design can reinitiate impingement jets at each stage, which greatly reduces the cross-flow impact and local thermal gradient. The staging of cooling air for the target surface also offers better utilization of the cooling capacity. Even by using 50% of the coolant designed for the single-stage impingement cooling, the multi-stage case can still sufficiently cool the target surface. The additional pressure loss penalty introduced in multi-stage design needs further efforts on design optimization.