Those moments are the rewards for active Bible reading. But achieving those moments requires a proper approach to reading the Bible. One approach is to read the Bible like a novel, beginning to end. Some people advise against that approach, but I think it's a useful way to gain a sense of the overall arc of God's plan. (However, if you wish to read the Bible from beginning to end, obtain a Bible specifically suited for that purpose. I really like this one.)
Another useful method for reading the Bible, especially those new to active Bible reading, is to memorize a verse that has some special meaning at that moment. A great place to start is the Psalms. For example, if you're seeking God's forgiveness, memorize Psalm 38:1 (ESV),
O Lord, rebuke me not in your anger,If you're asking God for strength, try Psalm 119:116 (ESV),
nor discipline me in your wrath.
Uphold me according to your promise, that I may live, and let me not be put to shame in my hope.If you simply wish to praise God, try Psalm 145:3 (ESV),
Great is the Lord, and greatly to be praised,These are simply, easy-to-memorize verses, and there are many others like them. Then, after having memorized the verse, recite it throughout the day whenever you have a free moment. By doing so, you'll soon find that you're coming close to praying without ceasing!
and his greatness is unsearchable.