In order to probe the order parameter symmetry of the heavy-fermion superconductor (HFS) CeCoIn<sub>5</sub>, we employ point-contact spectroscopy, where dynamic conductance spectra are taken from a nano-scale junction between a normal-metal (N) Au tip and a single crystal of CeCoIn<sub>5</sub>. The point-contact junction (PCJ) is formed on a single crystal surface with two crystallographic orientations, (001) and (110). Our conductance spectra, reproducibly obtained over wide ranges of temperature, constitute the cleanest data sets ever reported for HFSs. The point contacts are shown to be in the Sharvin limit, ensuring spectroscopic nature of the measured data. A signature for the emerging heavy-fermion liquid is evidenced by the development of the asymmetry in the background conductance, starting at <i>T</i>* (~ 45 K) and increasing with decreasing temperature down to <i>T<sub>c</sub></i> (2.3 K). Below <i>T<sub>c</sub></i>, an enhancement of the sub-gap conductance arising from Andreev reflection is observed, with the magnitude of ~ 13.3% and ~ 11.8% for the (001) and the (110) PCJ, respectively. These values are an order of magnitude smaller than those observed in conventional superconductors, but consistent with those in other HFSs. Our zero-bias conductance data for the (001) PCJ are best fit with the extended Blonder-Tinkham-Klapwijk model using the <i>d</i>-wave order parameter. The fit to the full conductance curve of the (001) PCJ at 400 mK indicates the strong coupling nature (2Δ/<i>k<sub>B</sub>T<sub>c</sub></i> = 4.64). However, our observed suppression of both the Andreev reflection signal and the energy gap indicates the failure of existing models. We provide possible directions for theoretical formulations of the electronic transport across an N/HFS interface in general, and the Au/CeCoIn<sub>5</sub> interface in particular. Several qualitative features observed in the (110) PCJ provide the first clear spectroscopic evidence for the d<sub>x</sub>2<sub>-y</sub>2 symmetry of the superconducting order parameter in CeCoIn<sub>5</sub>.