Similar to Which topics are essential for a crash course in QGIS?, which topics are fundamental to ArcGIS? Basic through advanced topics are welcome. Should be freely accessible to all. This is about ArcGIS because it seems better and easier than QGIS (I could be wrong and have to do more research, but no questions on this site has a basic answer to that).
I do not necessarily agree that ArcGIS is better or worse than QGIS--they each have their strengths and weaknesses. Given that, I often choose ArcGIS over QGIS for automating and customizing geoprocessing tasks. The automation topics that are fundamental for me in ArcGIS are:
I am also a fan of the spatial statistics tools in ArcGIS, primarily because it is really easy to work with spatial data for statistical purposes in a GIS rather than a programming environment such as R.
ArcGIS also has an incredibly powerful suite of tools called Spatial Analyst tools, which I believe is unrivaled.
You'd be best served to go through ESRI for some of their content specific training courses:
Depending on your personal needs, you'll certainly be able to find a course or courses that will help you out.
Some of the most useful topics to me are:
Other geoprocessing tasks
I think a different tack is appropriate as an answer to this question. Instead of focusing just on ArcGIS, or on QGIS, it might be more prudent to look at what these software packages are both used for. They are both similar in that they are used for
The links provided are quick overviews of geography and spatial analysis. The Python link is the main site for the programming language, and has many tips and code samples for implementing Python in all situations, not just GIS.
For more detailed information about these and other GIS related topics, this GISse question has many freely available GIS books: free sources - List of freely available GIS books.
The reason that I focus on these sorts of topics as opposed to simply the software, is that it is a tool used to implement these concepts.
The main free resource for learning how to use the ArcGIS suite of software itself, is the online help documentation:
The online versions of the documentation are going to have the most up-to-date information on any particular topic. In addition, they contain a vast number of code samples and examples of how to implement the functions and procedures using ArcPy, which is ESRI's python site package, or Modelbuilder.
It is up to the individual to decide whether ArcGIS or QGIS are better and/or easier. A lot of this depends on the process they are trying to run. One thing to remember based on your stipulation that the resources be Freely Available, is that ArcGIS is Proprietary Software. Please be clear, this is NOT a knock on the software, or on ESRI, just making it clear where they stand. They are a for profit company, which means that many of the resources related to the software, are not going to be freely available.
In many cases, your best bet for becoming familiar with ArcGIS, will be to take a class at a local community college or university. They tend to have intro classes, some of which may be available through an extension program, for lower cost. You may want to consider some sort of GIS Certificate program, which will provide not only the ArcGIS experience, but also some basic concepts in Geography and Spatial Analysis.
The other answers focus on the analytical side of GIS, which is important. It is also important to learn how to capture data and present it well.
Some important topics are:
These tools can seem complicated at first, but it is vital to have a good grasp of how they work. These topics are covered in Getting to Know ArcGIS with practical worked examples. It will also get you started with ArcToolbox and actual data analysis.