Quran was revealed to the Prophet (SAWW) both at once and verse by verse during 23 years. Firstly it revealed to the Prophet (SAWW) during 10 years of his prophet-hood verse by verse and again it was revealed at once.
The current arrange of verses of Quran is different of order of reveal of verses. Prophet himself ordered to change order of verses to make Quran as current order and it was by order of God. By this Change in verses of Quran, Quran became like a shuffled puzzle and more difficult to understand. When Imam Mahdi a.s. (the 12th Imam) come back of Occultation will order to change the sort of verses of Quran as its original revelation order.
In fact current arranged of Quran is by order of God to prophet SAWW.
All the Quran is revealed to Prophet Muhammad (SAWW) by angel Gabriel and the Prophet (SAWW) taught it to his companions and encouraged them to memorize it and thousands of companions of him memorized all the Quran under audit of prophet.
There is no story progression in start to end of Quran. But there are story progression for each story of Quran. Stories of Quran are also distributed in different Surahs.
There is no Jumping needed. Only:
لَّا يَمَسُّهُ إِلَّا الْمُطَهَّرُونَ
Which none shall touch but those who are clean.
Purify your heart of any evil moral and read Quran as a honest and serious truth seeker who is ready to pay any cost for finding truth. even if that cost is leaving wine, losing Job, being mocked by friends and family,... if so God himself opens door of your mind and heart to understand Quran so deeply that most of Muslims do not understand. God inspires the meanings to your heart and mind. Read Quran will deep asking of guidance from God himself.
If you prove to God that you do strive in way of finding truth, then God himself guides you and even if needed sends some of his Wali (close friends) to talk you and guide you without even you know him or he know you previously to you know God correct and can meet God.
وَالَّذِينَ جَاهَدُوا فِينَا لَنَهْدِيَنَّهُمْ سُبُلَنَا وَإِنَّ
اللَّـهَ لَمَعَ الْمُحْسِنِينَ
And those who strive in Our (cause),- We will certainly guide them to
our Paths: For verily Allah is with those who do right.
It is recommended to while reading Quran wash yourself and be clean and hold Quran in your hands with respect and it is better to read Quran when at least for 40 days no wine is drunk. Because God does not accept who drink wine until 40 days. Truth is expensive! And its price is higher than money. Its price accepting truth whenever comes to you at any cost and disobeying evil thinks coming in mind.
References and some suggested resources:
An Introduction to the Glorious Qur'an by Bahram Samii
The Qur'an in Islam
Understanding the Uniqueness of the Qur'an
A Reflection on the Collection of the Holy Qur’an