While notaddicted's reply can also be useful to you, I think the part of the puzzle that you are missing is "georeferencing". From your example, with a raster (bitmap, image) map of the US: the part where you start connecting that to lat/lon is georeferencing. Essentially, you want to tie a spatial reference to your image, so that the raster's state boundaries will correspond to state boundaries on a "real" map. You can think of it as pinning an image you know nothing about to a map you do know something about, and then copying the information from known to unknown.
There are plenty of ways to georeference an image. The easiest would be to use something like QGIS or GRASS (both opensource GIS software). You will need existing spatial data to base your (as of now, non-spatial) map on: a shapefile of the state boundaries would work great.
Once you've wrapped your head around georeferencing and the idea of spatial vs nonspatial data, you will probably get more out of the mapsfromscratch tutorial.
There are plenty of ways to georeference an image. The easiest would be to use something like QGIS or GRASS (both opensource GIS software). You will need existing spatial data to base your (as of now, non-spatial) map on: a shapefile of the state boundaries would work great.
Once you've wrapped your head around georeferencing and the idea of spatial vs nonspatial data, you will probably get more out of the mapsfromscratch tutorial.