How to start with STM32F103C8T6 board?
As it comes to datasheets they are indeed divided. "Reference manual" is complex descriptions of how to do things, configuration and detailed considerations about the whole MCU. "Datasheet" is on the other hand only a brief description of MCU features, pinouts, packages etc. There is also very helpful document for every STM32 line (F1,2...7) "Getting started with STM32F7xxxx MCU hardware development" which provides you with many informations if case you would like to design a custom board. 
In comparison to AVRs, registers descriptions are sometimes put a bit aside from main description of particular chapter.
STM32CubeMX is not a library, just a useful program from ST, which enables you to set pinouts, its configurations, set systems clocks etc and then, at the end, generate a code and the whole project. The project can be generated specially for Eclipse (System Workbench for STM32) in which you just import this project and you're ready to go.
The new library is HAL ("hardware abstraction layer") and try to use it. That's the new thing and sometimes it's hard to find examples on the Internet but don't teach your students outdated technologies. HAL is convenient to use and has documentation, so I think it's better choice.
Which library? Anyway it shouldn't be a problem to google them, I think.
Even if you would like to use only one OS (like Windows), Eclipse (AC6 = System workbench for STM32) is - I think - the best option. There is for example Keil's uVision 5 - but that's really terrible IDE - it lacks most of features IDE should offer (refractoring, click-and-search, proper error checking, finding references and many, many more). So, all in all, the Eclipse is pretty much the only reasonable choice for IDE for STM32.