Reading Quran while traveling is the highest form of "Dhikrullah" i.e. Remembrance of Allah.
However, there are certain precaution and etiquettes that should be remembered.
For driver- It is extremely important for the driver to focus entirely on driving the car. If reading of Quran does not cause any interference with driving then the driver may read the Quran. But if it cost any interference with driving than reading should be stopped as it may pose a danger to the life of driver.
If one is reading Quran from a Mus-haf and needs to touch the pages of Mus-haf then one has to be in the state of purity as well. Purity from minor Janabah is not required if one is reading Quran from memory.
One should always respect and honor the Quran by not letting it fall on the floor and keeping it at a higher place etc...
Passenger- The passenger may engage in reading Quran provided that he is in the state of purity from both forms of Janabah (minor and major) if he is to touch the Mus-haf of Quran. Otherwise, purity from minor impurity is sufficient.
One should not engage in conversations while reading Quran unless it is necessary.
If one has to engage in conversation then one should finish the recitation, close the Quran and then speak to others.
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America
7 Dhul-Qa'dah 1431/ October 16 2010