car mileage (odometer)

DVDdoug:
I really don't know how "modern cars" work and if you don't get an answer here you may have to try an automotive forum. ...Older cars had a flexible rotating shaft (a "speedometer cable") into the dash assembly where the rotation was converted mechanically or electrically to drive the speedometer and odometer.

I assume there is a speed sensor (probably coming off the transmission). You should get one or more pulses for every rotation of the drive shaft. You can get distance by counting pulses (and speed is calculated from that). You might have to experiment to get the pulses-per-kilometer, but once you can "catch" the pulses, programming the Arduino to count them is easy.

You might need your car's maintenance manual or search online to "find" the speed sensor and the connection in your wiring harness.
The automotive pulses/signals will be (nominally) 12V and that can fry the Arduino, so you'll need a couple of resistors to make a voltage divider, and I'd also recommend a "over-voltage protection" diode.
Graduation from what? What's your major?

thank you for your repaying ! Thank you so much!!
I think about way but I am new and I dont know much:(
let me tell you, If I design the Arduino to just calculate the distance? so I put the device I designed in the car, then send notification to the driver's iPhone when it is 5000 kilo?

it is right? if it is right can you help me how to do that?

Graduation from computer science bachelors..

thank you again!!