For many users, GIS means ESRI ArcGIS. While expensive in a commercial setting, they have rather generous educational licensing, including the provision of free copies to educators for distribution to students, one per licensed seat per year. I would advise at least ticking this box; I don't think people who learned GIS in other ways are less capable, but they might be less employable given the keyword-filtering resumes go through these days. The extensions to ArcGIS range from basic things that should be integrated into the main application to the amusingly archaic to essential tools for a certain niche.
The OSGeo stack is an obvious addition to this, but I don't think it's yet capable of being a full replacement in desktop GIS, at least not with the usability of Arc. Due to constant complaining about ESRI's annoyances, I tried to replace it for an entire summer with mostly QGIS, and failed. QGIS w/ plugins + GRASS + POSTGIS can be hacked around to achieve a lot of GIS functionality, I can believe that, but for learning GIS rapidly, I wouldn't recommend it. There are a lot of different projects under the OSGeo heading, though - in all likelihood you'll find use for some of them even if you don't touch the desktop functionality.
I'm always seeing MapInfo installations mentioned, but the one I used wasn't really mature / feature-ful in the same sense ArcGIS has been. The user interface was lacking, so perhaps it just hid the functionality from me.
Manifold has been highly regarded as an ArcGIS competitor that is at once commercially affordable, comprehensive, and extensively higher-performance than the sometimes antiquated ArcGIS code. They seem to be dragging their feet on updates & bug-fixes in the last few years, though. At the least, if ArcGIS fails to operate on extreme datasets, try this.
Remote sensing software is its own niche, with lots of features that aren't present in the ESRI stack. I've been exposed to ERDAS, and heard about ENVI and PCI. Those three constitute a majority share, but I'm aware that there are a decent number of options out there, some open source (I've heard good things about Opticks). In my own research area, 3D remote sensing is rapidly becoming a thing, as well - LIDAR and automated photogrammetry would be short topics in any RS course I'd teach. See: Meshlab, VSFM, & Photoscan.
For static-map cartography, you'll ideally want Adobe Illustrator, but there are several less expensive commercial choices that may suffice. My experience with Inkscape is a lot like my experience with QGIS - almost there, but missing crucial features like an effective layer dialog.
In the field of data manipulation, you'll definitely want to look at a thorough exploration of Python w/ LabPy. It's too versatile not to teach as a general tool at this point, there's the added inducement of ArcPy, and RPy adds the capability to use basically every statistical algorithm in the world. In addition, very big datasets are typically more amenable to a scripting environment than desktop GIS.
CAD & CAD-like GIS software, often used for surveyors / engineers, has a broad number of options led by Autocad which I'm not qualified to compare, but may not be necessary for a pure GIS program.
Client - Server
Hosting client-server stacks, which becomes important for some classes of GIS user, is the last niche I'm going to mention, but the requirements here are so varied as to make comparison difficult. Other than ArcSDE, The software here is often free, but the server setup and sysadmin to use it is not.
EDIT: I've had the opportunity to revisit QGIS recently and it does seem to have considerably improved, going from 1.5 -> 1.8, in stability & features.