Many research efforts have been directed towards enhancing the thermal properties of polymers, since they are classically regarded as thermal insulators. To this end, the present study focuses on the thermal investigation of poly(lactic acid) (PLA) filled with two types of carbon nanotubes (trade names: TNIMH4 and N7000), two type of graphene nanoplatelets (trade names: TNIGNP and TNGNP), or their appropriate combination. A significant increase in the thermal conductivity by 254% with respect to that of unfilled polymer was achieved in the best case by using 9 wt% TNIGNP, resulting from its favorable arrangement and the lower thermal boundary resistance between the two phases, matrix and filler. To theoretically assist the design of such advanced nanocomposites, Design of Experiments (DoE) and Response Surface Method (RSM) were employed, respectively, to obtain information on the conditioning effect of each filler loading on the thermal conductivity and to find an analytical relationship between them. The numerical results were compared with the experimental data in order to confirm the reliability of the prediction. Finally, a simulation study was carried out with Comsol Multiphysics® for a comparative study between two heat sinks based on pure PLA, and to determine the best thermally performing nanocomposite with a view towards potential use in heat transfer applications.