Spatially resolved profiles of the H alpha, [N II] lambda 6584, and [O III] lambda 5007 nebular emission lines, obtained with the Manchester echelle spectrometer combined with the 2.1 m San Pedro Martir telescope, have revealed the velocity structure of the nebular core and of one of the three (A, B, and C) inner "halos" of the high-excitation planetary nebula NGC 3242. The core is shown to have a cylindrical structure expanding at 25 km s(-1). The bright, diffuse, line-emitting, inner spherical halo A surrounding the intensely bright elliptical core is shown to be limb brightened, but its expansion velocity is unclear. The surrounding diffuse, 2.5 pc diameter halo B is modeled by a thick shell expanding at 20 km s(-1) although the contribution of scattered [O III] lambda 5007 emission is unknown at present. The origin of the broad profiles from the fainter, patchy, 0.44 pc diameter halo C is again somewhat uncertain and may originate in scattered core light. Far-infrared (FIR) observations with IRAS and the Infrared Space Observatory (ISO) reveal the presence of warm dust throughout the core and inner halos on NGC 3242. Alternatively, the split profiles could imply an expansion velocity of around 20 km s(-1) for halo C if scattering is discounted. There is a large filamentary structure of line-emitting gas 1.7 pc to the west of the nebular core. This is shown to be the western boundary of extended FIR emission from warm dust. A patchy component associated with this filamentary nebulosity emits the [O III] lambda 5007 line with anomalously high intensity yet is shown here to be kinematically relatively inert. Likewise, [N II] lambda 6584 profiles from the filamentary edge are very narrow with turbulent motions of 8 km s(-1). Certainly, the are is being photoionized by leakage radiation from NGC 3242, but the radial velocities of the line profiles are inconsistent with its origin as a simple shell expanding radially from this planetary nebula. An asymmetric lobe remains a possibility, in which case the are could be an ancient halo of NGC 3242. The more mundane possibility that the are is simply photoionized ambient gas is not completely ruled out by the present observations. Strange, faint, broad [O III] lambda 5007 line profiles are found over the whole 15' diameter area being considered here. The extended FIR emission region implies the presence of hot dust and suggests therefore that these [O III] lambda 5007 line profiles could have a scattered origin.