We present far-ultraviolet (1245-1680 Å), high-resolution ( R ~ 3 km s^-1 ) spectroscopic observations of three early-type stars (HD 32630, HD 158427, and HD 175191) located within 75 pc of the Sun recorded with the recently refurbished HST-STIS instrument. These data have allowed an absorption study of the local interstellar gas to investigate some of its physical, chemical, and kinematic properties. A local stellar continuum level was assigned to each of the various interstellar absorption lines, and the resultant residual-intensity absorption profiles were then fit with a theoretical multicomponent model of interstellar cloud component velocity and cloud component column density. These model parameters were then used to investigate the physical and chemical states of the local gas along each of the three interstellar sight lines. Eleven velocity components were required to fit all of the observed absorption line profiles, and we have identified the source of most of this absorption as arising in gas associated with the Local Interstellar Cloud (LIC) complex distance of < 10 pc . The remaining absorption path through the Local Cavity (over the distance 10 to 65 pc from the Sun) is mainly deficient in these diffuse cloudlets. We have observed a wide range of temperatures ( 102 to 105 K ) and ionization states (5 to 50 eV) of gas within the Local Cavity. Highly (photo) ionized C IV absorbing gas with a column density of log N ( CIV ) = 12.20 ± 0.17 cm^-2 has been detected at the velocity of the G cloud within the LIC complex along the sight line to HD 158427. However, no highly ionized gas was detected along the other two sight lines toward HD 175191 and HD 158427, neither of which exhibited absorption in any line species at the projected velocity of the G cloud. The nondetection of local highly ionized gas along these two directions could be due to variations in the levels of photoionization found throughout the Local Cavity, which may well be linked to the geometry of the local clouds which are very filamentary and may be highly contorted such that local shielding may play a far larger effect than previously imagined. Based on the column-density ratios of the interstellar C II and C II* absorption lines, we have derived an electron density of n_e = 0.09 ± 0.06 cm^-3 for the Hyades cloud seen toward HD 32630. Estimates of n_e for other local clouds are less reliable due to the large errors associated with the column density derived from their saturated C II line profiles. Even so, our range of 0.05 < n_e < 1.1 cm^-3 estimated for these other local clouds is consistent with previous determinations. Element abundances (relative to that of S) have been derived for several local gas clouds, such that Al, Fe, and Ni are depleted generally by a factor of ~ 10 compared with their solar values. However, large variations in the element depletion values are seen between different cloud components. These abundance determinations can be reconciled with previous estimates for the local gas if, instead of using individual cloud component column-density values, a single absorbing cloud model is assumed for the total integrated sight-line column-density values. Due to ultraviolet line-saturation problems associated with studying absorption of the interstellar gas residing in the 30-80 pc distance range from the Sun, future studies may require access to other (intrinsically weaker) line species recorded at high spectral resolution in the far-ultraviolet (912-1170 Å) wavelength region.