Creating maps with GIS
Though the methods for creating distribution maps using GIS are fairly straightforward there is an absence of practical guides on the subject.
Accordingly I began to write a series of tutorials on this subject, downloadable here. It is an incompleted draft from a couple of years ago, hopefully all the links have persisted.
First steps in GIS
There are introductory GIS courses available, notably those by Field Studies Council such as one by Rich Burkmar. The scope will just be the British Isles though, I guess.
If you wish to explore GIS at no cost (all the resources you need are free) then the first step would be to download QGIS and take a look around the interface. Second step is to set up your "housekeeping", organise the folders on your hard disk ready to receive all the free data you'll have to download. Keep the downloaded maps separately from any project work.
Now see if my tutorials make sense.
The links to the main TDWG data (Country outlines for Europe) has changed since I wrote the tutorials, Try https://github.com/tdwg/wgsrpd on their page World Geographical Scheme for Recording Plant Distributions (WGSRPD)