Depending on your image type you can use Esri Raster Mosaic dataset - have your tiles and mosaic too without any (much) extra storage.
The mosaic dataset works like a VRT file where the rasters are referenced but not imported so the table in the database stores the path and location of the tiles; overviews are a good idea but they don't take up much space. For the purposes of geoprocessing you can select the tile or the mosaic dataset; all raster geoprocessing tools will use the mosaic dataset as a single raster.
As for actually creating it, consider a Terrain which you can load all sorts of elevation source data, even to a very large size. Then using your output extent setting on your tools (and snap raster as well, just to keep it neat) you can export tiles, more than one at a time, as raster.
For GIS software that's not Esri you can also have the same tiles as a VRT raster, make it using GDALBuildVRT. This is also an Esri raster type but is significantly slower than a mosaic dataset (notably, many times slower).
Hillshades help visualize terrain, which can be created in Esri using spatial analyst. Should you not have spatial analyst or not like the Esri hillshade (it's a visual preference) then you can use GDALDEM to create one, or tiles and mosaic that too, the DEM must be in a GDAL format (sorry, mosaic dataset isn't a GDAL format). You can use a VRT of the tiles, however this format has proven to be very slow on large datasets, best to use ERDAS Imagine (.img) or GeoTIFF tiles (with some overlap) and hillshade each one then clip the overlap area (GDAL_Translate -projwin or Esri Clip_management) and build a hillshade mosaic dataset.
If you aren't comfortable with command line, you can do a hillshade from QGIS in the geotoolbox: