Isaiah 65 discusses two types of people - those who rebel against God and those who listen to Him. This chapter also describes what will happen to the rebellious and the faithful. The faithful will inherit the earth and all the blessings that comes with it.
Isaiah 65:2 describes in which way the rebellious are not faithful. It says that they walk in a way that is not good and they walk after their own thoughts. In other words, the rebellious do not seek nor follow the will of the Lord. They have their own agenda.
Isaiah 65:3-4 suggests that the rebellious worship idols (sacrifice in gardens and burn incense on alters). They also seek after spirits of the dead (remain among graves). Idol worship is not a thing of the past. Many people sin in idolatry today. Just because we don't pray to statues doesn't mean we are not idolaters. Today, many suffer idolatry in the form of worshipping the self-image. I've referred many times before to a fascinating and scathing talk given by Spencer W. Kimball. That talk was relevant when he delivered it and it remains relevant today. Read Spencer W. Kimball, “The False Gods We Worship,” Ensign, Jun 1976, 3.
The rebellious are not necessarily people who are not members of the LDS church. Isaiah 65:11 suggests that those who forget the Lord's holy mountain, which is the temple, are also numbered among the rebellious. In the temple we make covenants with the Lord. The faithful are those who make and keep covenants with the Lord.
In Isaiah 65:13-14 we learn the faithful will eat, drink, rejoice and "shall sing for joy of heart."
New Heavens and Earth
Isaiah 65:17 reads, "For, behold, I create new heavens and a new earth: and the former shall not be remembered nor come into mind."
The Tenth Article of Faith teaches us that we believe "the earth will be renewed and receive its paradisiacal glory." Just as the earth was baptized at the flood and was "renewed" so too it will be baptized by fire and be renewed. Many of the references of a new earth are found in Isaiah (see Topical Guide: Earth, Renewal of and Isaiah 66:22-23).
As I hinted at above, the rebellious will not be a part of the new earth. The faithful who have made and kept covenants with the Lord will receive this new earth. Isaiah 65:20 teaches us (as the Vulcan cliché goes) that everyone will live long and prosper. Babies and children will live a full and long life. Women and men will live a full life and we'll live as long as trees (perhaps as long as sequoias).
The faithful will labor and build homes and grow gardens and enjoy the fruits of their labors without having to worry about enemies destroying them (Isaiah 65:23).
The earth will be full of the knowledge of the Lord and even the animals will not injure one another (Isaiah 65:25).