What is the biblical basis for the belief that God likes or feels affection toward us? To some people, and even to Christians at times, it seems difficult to believe that the all-powerful Creator of the Cosmos likes or feels affection for humans personally. This was the topic of a recent question addressed to the Catholic perspective. I would specifically like to ask: what is the biblical basis for the belief that God's love for people includes a genuine individual affection?
John 3:16:


  For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.


Romans 8:16-17:


  The Spirit itself beareth witness with our spirit, that we are the children of God:
  And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together.
  For I reckon that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us.


There are also many, many instances of angelic messengers and servants of God being sent to serve, protect, and teach people, identified as individuals, and not just as groups or important leaders. See Genesis 31; 1 Kings 17; and Daniel 3 for a few.

Aside from the examples above, the New Testament is also replete with examples of God's love for us as His children. The recorded miracles performed by Jesus Christ were almost always done for individuals of apparently little or no great fame or reputation. The Messiah, sent to save all mankind, frequently spent His precious, limited time ministering to individuals.

Your question is specifically about the Bible, but I'd also like to add that the Book of Mormon has many plain, beautiful examples and explanations of the love of God. 2 Nephi 2-4, Enos 1, and Mosiah 27 contain a few of my favorites.