How can I authenticate/read/write/access data on my card with my SpringCard reader?

There is no easy answer: this fully depends on your card and your SpringCard reader. You should first read the documentation of your card, then you should download and read the developer's reference manual, or the developer's guide of your product, from our main website. This should give you all the information you need.
If you don't know how to start, you can check this page in our blog. It explains in detail how to read and write the content of a Mifare Ultralight card, with a Prox'N'Roll PC/SC: having read the above-mentioned documentation, you should be able to easily adapt those steps to your specific situation.

How can I know if my reader is a Prox'N'Roll RFID Scanner, or a Prox'N'Roll PC/SC?

The first thing to do is to look at the sticker for “PC/SC” or “RFID Scanner”. This should give you the right answer.

Now, if you don't have a sticker anymore, or if it is unreadable, or if you're still unsure: open a text editor (notepad on Windows for example), and then put a card on your reader: if something is printed, then you have a RFID Scanner.