Tajweed and its application can only be learned with a qualified teacher.
The rules themselves can be studied independently, but their correct application can only be done by listening to, reciting to, and being corrected by, a qualified teacher of the Qur'an.
Do all Arabs know how to recite the Qur'an with proper tajweed?
Unfortunately many do not know proper tajweed. In the days of the Prophet, peace and blessing upon him, there was no need for the study of tajweed because they talked with what is now known as tajweed; in other words, it was natural for them. Now, over 14 centuries later, colloquial Arabic has changed radically from the classical Arabic with which the Qur'an was revealed, and Arabs have to study tajweed, just in the same way that non-Arabs do.
I want to learn to recite the Qur'an with proper tajweed, but don't know where to start.
The first step is finding a qualified Qur'an teacher who will listen to you and point out your mistakes, and help you practice fixing them. You will need to learn the Arabic letters and vowels too.
What if there aren't any qualified teachers where I live?
Then your task is going to be a little harder, but certainly not impossible. You need to work with tapes of good reciters, for example, Sheikh Abdullah Basfar, or Sheikh Mohammed Hosary. There are sets available on the Internet and in Islamic stores that are called "mu'alm". In these sets, the reciter reads a phrase, and then there is a blank space on the tape, allowing the student to repeat what the reciter has just read.
You will need to work on the pronunciation of letters by getting a description of the articulation points and practice placing your tongue, lips, or finding the place in your throat where the letter is articulated from, and compare it to the letter when recited by the Sheikh.