How to read data from a 3.5mm headphone jack?
Analyzing audio with a Arduino isn't very simple for someone that just
  wants to get data from a TI-84, and other devices, via a 3.5mm jack.


If you just want to communicate with a device like a TI-84 that uses a 3.5mm jack as just a connector for some type of serial interface, this shouldn't be too hard (although most of the helpful links online are dead.)

I don't know much about the TI-84 protocol, but since you wanted a basic tutorial, I'm guessing you may need some help with Arduino serial communication. Arduino Serial is a great place to start, and LadyAda has a good serial introduction.

Here is an image that shows you to make the connection, (I'm not sure if any level converters are needed, but you can read the link below.)



And the code and discussion can be found here: Arduino to TI Calculator Linking Routines


  PS: I don't want to use a modem like this, I to learn and make it not
  communicate only with iOS/Android devices.


I think the board you quoted is basically a level converter, converting 3.3volt signal to 5volt arduino, and vice versa. Also it said it was a 4 conductor cable so it is different than the TI-84 cable, I believe the TI's is only a 3 conductor.

Links:


Video: Arduino to TI-84+ Linking
Video: TI-84+ to Arduino communication
Slightly complex discussion
Less helpful discussion, but with code example
A somewhat not relevant, but very cool project, Expand TI 84+ calculator with touch screen