What GIS software is required to export a GIS map as a hi-res raster image?

Did you try Qgis? I think wil cover you. You wil find a lot of help on Internet. It is very easy to manipulate the data. If you want to change their style or icons it is very easy. And of course Tiff export is a standard. Is open source and cross platform, that mean you can install it in any Opereting System.

More info: From here

Download: From here [Standalone Installer (recommended for new users)]