Here’s my understanding (if I’m wrong about something I’m sure someone will correct me):
At the moment of death our moral state and our eternal destiny (heaven or hell) will be determined for all eternity. This will depend on whether we are in a state of mortal sin, which means that by some act we have rejected God in some serious way and have not then repented of it. (for Catholics repentance also includes the resolution to go to confession as soon as possible) The Resurrection of the dead and our ability to enter heaven is indeed due to Christ’s sacrifice on the cross, since only the Son of God could offer fitting reparation to God for our sins.
After death there is an immediate judgement for each person in which their eternal destiny is determined. If in a state of mortal sin they go to hell, which is eternal separation from God and complete unhappiness. If not in a state of mortal sin, they will go to heaven, though they may have to go first to purgatory if they have some venial (i.e. not so serious) sins for which they have not made retribution on earth. (for clarification, the afterlife is outside of time, so what kind and amount of “time” they will have to spend in purgatory is unclear)
Purgatory is a place of purification in which the soul will be washed clean of the stain of sin and made ready for Heaven. This will involve some sort of pain, but it will be a joyful and hopeful place nevertheless because the souls will be assured of salvation.
Heaven is the state of perfect happiness and communion with God, where we will see God face to face. It will last forever and be the culmination of all goodness.
We will not be reunited with our bodies after the personal judgment, but only after the final judgment which will occur at the end of the world. After this final judgment there will be “a new Heaven and a new earth” (im not sure, but i think that the “new heaven” refers in this case to the physical heavens, i. e. the universe) and we will live both in our glorified physical bodies and in heaven, in a perfect union of body and soul. Our bodies will not be merely physical bodies, but also spiritual bodies, not bound to the restrictions of the physical world and therefore perfect reflections of our souls. The damned will also be reunited with their bodies, which will be resurrected and immortal, and they will be both body and soul in Hell for eternity.
The New Jerusalem does indeed refer to Heaven, though whether it also points to a city in the new earth is more than I could tell you
Hope this helps, I’m no theology major but I believe I put the Catholic position accurately.