You also want to think about what your real interest is. Do you want to be a GIS developer versus a analyst in a particular field. You will see many places want to have a GIS person that has skill-sets in development as well as GIS.
But you can be a strong GIS user if you study Landscape Architecture, Civil Engineering, or even just starting as a CADD Analyst like me.
There are a lot of good generalist out there in the industry; who can just setup a basic spatial DB, slap a web-ui on a OpenLayers page consuming Bing/Google and be happy with that. Where you see the people who are most passionate about GIS is the ones who have it just as a tool in there box. Knowing your data domain is a great place to grow, to know what you want to build and support to be able to anticipate needs.
Core Hardware and Software are important in many areas, because if you can't see how you systems need to grow, then you won't be able to sustain the efforts.
What do you really want to do?