The basic issue is that I have shapefiles saved in WGS84/Pseudo Mercator (EPSG: 3857) and some DEM data saved in NAD83/UTM Zone 10N (ESPG: 26910), but can't get them to line up.
I'm using QGIS and have my file CRS set to WGS84 (3857) with on-the-fly projections enabled. My global settings allow on-the-fly to automatically be applied to any new layers added that aren't in the drawing CRS.
The DEM files that I'm having trouble with are from the National Elevation Dataset (NED) and were obtained through the NRCS Geospatial Data Gateway. I've tried to change the DEM layer CRS in the layer properties. I've also tried to change the CRS by right-clicking the layer and choosing "Set Layer CRS" and I've tried to "save as..." and alter the CRS for a new layer file. None of those have worked.
Then I tried to use GDALWARP to reproject the DEM from NAD83 to WGS86, but the DEM just ends up in exactly the same spot.
Trying to change the CRS of the drawing and the shapefiles to meet the DEM is similarly counter productive. They are outside the bounds of the NAD83 UTM Zone 10N, so they just disappear.
I'm pretty new to QGIS and GIS in general, so it's likely that I'm making a real rookie mistake here. I'm using QGIS 2.8.1 Wien on a Windows 8 computer if that helps.