The ubiquity of GPS-enabled mobile phones and applications such as Google Maps have made location based services mainstream, whereas not so long ago, GIS was mostly carried out on very specialized software in a desktop environment. At the same time, there has been an explosion in data collection and analysis (so-called Big Data), a large part of which could be considered geographical in nature.
So, while I totally agree with @underdark, I would add a couple of other areas which I think would/could be useful for someone setting out on a GIS career.
1). Databases. The Postgres/Postgis combination provides an extremely powerful combination of a very stable and mature RDBMS, with hundreds of functions, allowing for the analysis of vectors, rasters and topologies. Dealing with data where it is stored, is often much more performant and scalable than doing it in desktop GIS.
2). R (or some other statistical language). There are a lot of very good spatial statistics libraries for R and a lot of GIS work involves spatial statistics (this also fits well with the growing area of Data Science/Big Data). Python is growing in popularity for statistics also, but R is probably still the leader.
4). Very popular in the GIS world and a scripting language for both ArcGIS and QGIS. There is also a very nice web framework, Django, and it's extension, GeoDjango.
While I have an MSc in GIS (also done for a career change), which gave me a really good grounding, I really wish I had done some introductory algorithms and data structures classes, as these really helped me to see GIS problems in a different way. You can, of course, pursue a GIS career without programming, but it will definitely help a lot, if you have some. Good luck, whatever you decide.