Hi John,
GaiaGPS on Android supports loading local maps from MBTiles files, that contain jpeg or PNG images. They can be imported throughout the main menu on the left side of the app, and they will then show in the map sources menu.
MBTiles are always in SpericalMercator projection, EPSG:90013 or EPSG:3857.
We tend to work with free and open source tools for data processing, so the rest of my suggestions are focused on that, but if you are used to ESRI tools, you should be able to use those to generate MBTIles.
There are a couple options for converting data to MBTiles, TileMill is a good option, free, and relatively easy to use, check out https://www.mapbox.com/tilemill/docs/guides/reprojecting-geotiff/
If you are comfortable on the command line, gdal, gdal2tiles, and mb-util can get the job done, and is my preferred toolset.
Another option would be to use mapTiler, a commercial product http://www.maptiler.org/ with a free version available.
If the data your working with is elevation data, you probably want to render it into hill shading, contours, or a hypsometric map. Mapbox has a nice guide on working with terrain data https://www.mapbox.com/tilemill/docs/guides/terrain-data/
Shapefiles can not be imported directly, if the data is only lines and points you could convert it to KML or GPX. Currently only lines and points are supported, so if you are looking to show polygons, you would best rendering the shapefile into a raster layer, putting it into an mbtiles file, and importing that.
Happy Hiking,
Jesse Crocker