Allah Most High has said, "We send down in the Qur'an that which is a healing and mercy to those who believe." (Qur'an; Surah al-Isra')

The benefits and rewards for learning the Qur'an are many.  You can find these rewards mentioned in several Qur'anic passages and hadith texts. Just to name a few:

Are they the same, those who know and those who know not? – (Qur'an; Surah al-Zumar)

Allah raised the believers and those whom He grants knowledge lofty degrees – (Qur'an; Surah al-Hadid)

The best of you are those who learn the Qur`an and those who teach it – (Sahih Bukhari)

No people gather in a house of the houses of Allah to read the Qur`an and study it between themselves except that tranquility descends upon them, mercy covers them, angels shield them, and Allah makes mention of them to those in His presence – (Sahih Muslim)
 
In addition, there are many virtues for certain chapters (suwar) and verses (ayat) of the Qur'an.  Of them, the Prophet (May Allah Bless him and Give him Peace [this should be replaced with the Arabic text for the saying as it is much shorter and nicer]) has mentioned:

1. al-Fatihah (01):  "In Surah al-Fatihah there is balm for all sickness."  He also said, "Shall I tell you of the surah which is the greatest?  It is Surah al-Hamd, which has seven verses called the Saba' Mathani (the Seven Oft-repeated Verses) and are the representatives of the entire Qur'an."
2. Ya Sin (36):  "Whoever reads Ya Sin solely for the pleasure of Allah Most High will have all his earlier sins forgiven.  Therefore make a practice of reading Ya Sin over your dead."
3. al-Waqi'ah (56):  The one who reads al-Waqi'ah every night – poverty will not befall him."  He also said, "Surau-l-Waqi'ah is Suratu-l-Ghina (the Surah of Wealth), so read it and teach it to your children.
4. as-Sajdah (32) and al-Mulk (67):  "For the person who reads these two chapters between Maghrib and 'Isha [the reward] will be as though he has stayed awake during Laylatu-l-Qadr."
5. ad-Dukhan (44):  "Whoever recites ad-Dukhan on the night of al-Jum'ah, seventy thousand angels will ask for forgiveness for that person until morning."