Allah promises in al-Qiyamah 75:19 that He explains the Qur'an.

Of course this does not mean anyone can just open the Qur'an and gain some sort of instant mystical wisdom. We have to follow the methods the Creator of the Qur'an lays out for us. 

And the Qur'an is the only book ever that teaches the reader how to understand itself...

BEFORE WE START, we seek refuge with Allah from satan the rejected. We do this before we read the Qur'an or take any of the following actions. This means not just saying the words of seeking refuge: we actually clear our minds of all other ideologies or agendas and seek refuge with Allah to understand His Book as Muslims for Him.

1. Read (iqra). 

We need to regularly READ the Qur'an. 

This habit molds our thinking to see the world as He would have us see it.

2. Consider it with care. 

Reading/reciting with no understanding is NOT enough. 

Reading with an agenda, or preconceived notions, will dim our vision.

3. Pray. 

Perform salaat as called for AND make dua for increased understanding. 

4. Convey it to others. 

Invite to the teachings in the Qur'an, 

openly or subtly, one ayat at a time,

or even just one idea at a time.

5. Practice the teachings. 

There are too many invitations to do good to list here. 
Find them. Do them. 
Primarily: Remember Allah Often and Help Others. 

6. Consult with others

Do not assume you or they have any final 'correct' understanding; 

and we must listen with open-minds while avoiding blind acceptance;

this must be a continuous and on-going process;

 

7. Be patient. 

Allah will manifest His will in His time. 

Trust him.

There are ayah to support each of the above habits; and this is still a work in progress.

(As it will likely ever be.)

Let us also always maintain Habit Zero: Self-Examination.  

The early Muslims, particularly in the second generation of the Islamic era, were fond of examining themselves critically. Before going to bed at night, each person would try and recall his offensive action, thoughts, utterances and dealings for the day, then analyze them and identify their own moral faults.

We find this process of looking at ourselves critically to be a necessary tool in all arenas of growth; it is not limited to learning to understand the Qur'an or to dealing with our fellow Muslims.