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Ensure you have ArcTutor installed (comes with ArcGIS Desktop install)
Start doing the tutorials.
Id also advise paying a trip to the ESRI Virtual Campus.
- Plenty of free training on there to get you started.
Go to video.esri.com for inspiration.
This question has been converted to Community Wiki and wiki locked
because it is an example of a question that seeks a list of answers
and appears to be popular enough to protect it from closure. It
should be treated as a special case and should not be viewed as the
type of question that is encouraged on this, or any Stack Exchange
site, but if you wish to contribute more content to it then feel free
to do so by editing this answer.
I'd like to suggest two books. The first one is Learn to Think Spatially from the National Academies Press.
It is actually about the nature and functions of spatial thinking and shows how spatial thinking can be supported across the K-12 curriculum through the development of appropriate support systems like GIS but it should give you an idea on how to include spatial thinking in your classes.
The second book is Practical GIS Analysis.
It is practical guide for solving geo-spatial problems independent of specific GIS software and hardware. It focuses on how GIS tools work, and how you can use them to solve problems in both vector and grid GIS worlds. It teaches the basic GIS operations like overlay, intersect, etc and how you can combine those together instead of teaching you about specific GIS software packages' tools.
It also includes real-life applications from urban problems including real estate query, irrigation analysis, urban emergency response, address geocoding, street management, resource allocation, groundwater analysis, auto accident analysis, parcel analysis, and optimal path analysis. There are also more than eighty GIS problems (and solutions) which should help you test problem-solving abilities. This should make a good starting point for your classes. You can view the table of contents here.
Hopefully those two books should help you achieve. I'm glad your school has this initiative. For so long the focus has been on specific tools, to the point that students equate GIS to ArcGIS when it is so much more.
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I'd suggest starting from http://www.w3.org/Mobile/posdep/GMLIntroduction.html.
If you're planning to work with application schemes in INSPIRE, you'll also need a thorough understanding of XML and XSLT (Extensible Stylesheet Language Transformations).
I think the task doesn't require in-depth understanding of UML. You'll have to be able to read it. Anyway, you can find tons of resources at http://www.uml.org/.
You may be able to glean some general info about INSPIRE from some of these links: