Awesome thread!
Cultivating intimacy with God.
One should set aside time for prayer, meditation, and praying in the Spirit. Also, as many said here, learning to follow the leading of the Spirit.
I'd like to add something else...
There is also the cultivation of an ontological intimacy. Intimacy with God is more than just a prayer life. It's more than laying petitions, supplications, requests, praise, and worship before God. It is more than feeling emotional towards God or sensing emotion from God. True "spiritual intimacy" with God is "oneness". This is a cultivation of the "oneness" of spirit through the Holy Spirit that indwells the spirit of the born again believer.
The believer who lives a life of spiritual intimacy with God has realized and experienced this union with the LORD. God doesn't desire that we live lives wherein we are merely looking up and serving a God who is somewhere "up there". No, God desires to indwell us and to be found in the deepest recesses of our beings. Through this union/oneness we cultivate the "mind of Christ" by renewing our minds through study of the Word and tapping into the very mind and thoughts of Jesus. We are to "put on Christ", allowing ourselves to be conformed into His very image and likeness. This is more than emulation. In fact, it is allowing His Spirit to live His own life, will, desires, and plans out through us. We are to live as "living extensions" of Him in this lost and darkened world; He being the True Vine and we being the branches. A vine and its branches are a single organism. His life, Spirit, and power will flow through this intimacy and oneness as the life, spirit, and power flows through the vine into it's branches. And in this union, we will bear much fruit. In this union we see the unseen, hear the unheard, and manifest the unknown... through Him who dwells within us, within our very being, within our "inner man". This is true "intimacy with God". Oneness of the mind and spirit with the Holy, the Eternal, the Source, the Creator, the Alpha, the Omega, the First, the Last... the great... I AM. We become "one spirit" with the LORD Himself.
In this union He partakes in all that we are... and we partake in the divine nature, all that He is, was, and ever will be. And it is all available to us through grace.
Here are some spiritual mysteries to ponder from the Scriptures...
John 14:20 (ESV)
20 In that day you will know that I am in my Father, and you in me, and I in you.
John 14:23(ESV)
23 Jesus answered him, “If anyone loves me, he will keep my word, and my Father will love him, and we will come to him and make our home with him.
John 15:1-7 (ESV)
15 “I am the true vine, and my Father is the vinedresser. 2 Every branch in me that does not bear fruit he takes away, and every branch that does bear fruit he prunes, that it may bear more fruit. 3 Already you are clean because of the word that I have spoken to you. 4 Abide in me, and I in you. As the branch cannot bear fruit by itself, unless it abides in the vine, neither can you, unless you abide in me. 5 I am the vine; you are the branches. Whoever abides in me and I in him, he it is that bears much fruit, for apart from me you can do nothing. 6 If anyone does not abide in me he is thrown away like a branch and withers; and the branches are gathered, thrown into the fire, and burned. 7 If you abide in me, and my words abide in you, ask whatever you wish, and it will be done for you.
Romans 8:29 (ESV)
For those whom he foreknew he also predestined to be conformed to the image of his Son, in order that he might be the firstborn among many brothers
Colossians 1:27 (ESV)
To them God chose to make known how great among the Gentiles are the riches of the glory of this mystery, which is Christ in you, the hope of glory.
Galatians 2:20 (ESV)
I have been crucified with Christ. It is no longer I who live, but Christ who lives in me. And the life I now live in the flesh I live by faith in the Son of God, who loved me and gave himself for me.
2 Peter 1:3-4 (ESV)
3 His divine power has granted to us all things that pertain to life and godliness, through the knowledge of him who called us to his own glory and excellence, 4 by which he has granted to us his precious and very great promises, so that through them you may become partakers of the divine nature, having escaped from the corruption that is in the world because of sinful desire.
1 Corinthians 6:17 (ESV)
But he who is joined to the Lord becomes one spirit with him.
Below, I have shared a chart expressing this spiritual reality. This is "intimacy with God".