TFS=Team Foundation Server. All things microsoft that require storage and integration get run through this. VS=Visual Studio. All things that are "visualized" are pushed through this. MTM=Microsoft Test manager. This manages all tests.
So with this then TFS stores everything and synchronizes it. VS is a method to visualize things in order to interact with them (i.e. code files primarily). And MTM is for the test execution/creation/updates/reporting. Versioning is important as these all correspond with one another. 2013 means you need to get the 2013 version of MTM and VS as well since you want all of your functionality to synch up correctly. If you don't then there may be some problems with the way some "work items" function.
As a simple note "work items" are anything that is stored in TFS and work is done against. So test cases are "work items" that have a specific type of "test case". Just as bugs are "work items" that have a specific type of "bug". These can be custom defined as well so it depends on what your company has implemented.
Review 2012 ones as those will probably give you the bulk of what you need to know and have been around for awhile. 2013 is minor tweaks and modifications that you can google/youtube specifically. I have found the best ones tend to be on youtube...
~Google/Youtube "TFS tutorial" and "MTM tutorial" and "Visual Studio tutorial"
You will find some helpful tutorials starting from beginners up. Most of TFS and Visual Studio are focused on coding so unless you need to utilize source control and the code editor then a basic knowledge is good enough.
Web and stand alone can accomplish most of the same things, but the web looks prettier and is easier to pick up. Some things are not supported through the web though so you will want to look at it through the stand alone.
MTM and Visual Studio access the same "Work Items" in TFS and therefore you can see the test cases from both...but only MTM has the specific functionality to create/update/execute tests as you will want to do.