The ultimate cheat to avoid confusion about projections is to go to your local thrift store and buy a used globe. They're usually pretty cheap, at least around here. Then spray paint it flat white and draw on its surface to rough things out. From there, you can stitch together and reproject a few pictures to get an image in something like the Equirectangular projection.

If you're intent on doing a digital-only map, try using a template like the one you referenced above. Load it into Photoshop and resize as appropriate. Then make a new layer and draw onto that. Once in a while, use a program that supports reprojecting images like G.Projector to see how your map looks in other projections. Drawing in the distortions digitally can be a bit confusing because of how the grid lines converge toward the poles.

Ideally, you'd use a paint program that has an understanding of projections, but I haven't run across a good and easy-to-use one yet. If I'd ever get things in gear and add projective brushes to Fractal Terrains, it would be close to suitable for this use. As it is, FT does make viewing things in various projections a little easier, if not exactly correct.

As far as reprojecting your flat map into a world map, it's a matter of what you already have. If your continents are reasonably equatorial, you can probably just select an existing projection like Mollweide or Hammer and declare that your map is defined in that projection (any distortions around the edge are likely to be far away from the primary area of interest along the central meridian and you can simply say that they are due to cartographic error).