[Abridged] We present Gemini-N GMOS-IFU observations of the central starburst clumps and inner wind of M82, together with WIYN DensePak IFU observations of the inner 2x0.9kpc of the disk. These cover the emission lines of H$α$, [NII], [SII], and [SIII]. We were able to accurately decompose the emission line profiles into multiple narrow components (FWHM~30-130kms) superimposed on a broad (FWHM 150-350kms) feature. This paper is the first of a series examining the optical structure of M82's disk and inner wind; here we focus on the ionized gaseous and stellar dynamics and present maps of the relevant emission line properties. Our observations show that ionized gas in the starburst core of M82 is dynamically complex. Localised line splitting of up to 100kms in the narrow component is associated with expanding shells of compressed, cool, photoionized gas. We have been able to associate some of this inner-wind gas with a distinct outflow channel characterised by its dynamics and gas density patterns, and we discuss the consequences of this discovery in terms of the developing wind outflow. The broad optical emission line component is observed to become increasingly important moving outward along the outflow channel, and in general with increasing height above/below the plane. Following our recent work on the origins of this component, we associate it with turbulent gas in wind-clump interface layers and hence sites of mass loading, meaning that the turbulent mixing of cooler gas into the outflowing hot gas must become increasingly important with height, and provides powerful direct evidence for the existence of mass-loading over a large, spatially extended area. ; Comment: 36 pages, 21 figures, accepted for publication in ApJ