I think the common thread that ties verses 1-6 to 7-13 is the glory of God. Not only is His glory revealed in nature, His creation, but also in His written Word. Just as creation inspires awe and wonder at His creativity and might, God's Word inspires that same awe and wonder, especially when we see how much He, a powerful and holy God, loves us.
The psalmist seems to bask in God's Word by continuing in his observations of what the Word does in the lives of humans. He doesn't just say something like, "God's Word is great and does a lot of good for those that read it." He gets specific. It is obvious that the psalmist has experienced the reviving, wisdom-creating, joy bringing, insight giving power of the Word. I definitely feel as the psalmist does about the Word. Once you really dig in to the Word and are given guidance by the Holy Spirit, the Word comes alive and changes you from the inside out.
The psalmist's wonder in creation seems to inspire him to think of the glory of the Lord and His Word. When he looks at creation, he sees the Lord. I think there are so many things that the Lord has created for us that we take for granted. In not seeing the Lord in what He has created, many beautiful things have been corrupted.
In the prayer that closes Psalm 19, the psalmist is asking God to continue His work in him in order for the Lord to be shown to others through the psalmist. In the words of the New Testament, the psalmist is asking the Lord to help him die to self and be a living sacrifice, giving up his worldly thoughts for the higher thoughts of the Lord. I came across this prayer in an earlier reading of Psalms and made that a prayer. It definitely helps you to keep your priorities straight.