What does the Catholic Church say happens on the New Earth? There is a tiny blurb in my text book saying that 


  Even the earth will be transformed in some way


referencing revelation 21:1-3 which it goes on to quote, but not actually explain. 

So what happens on a new earth? Does it stay on the new earth, is that where folks who live through the second coming hang out until they ascend into heaven?
Creation will be renewed one day, freed from corruption and illumined by God’s glory. - (RSVCE) note for Rev 21:1.




New Advent says the following on the new heaven and the new earth:


[For the the elect], after the last judgment the new heaven and the new earth will afford them manifold enjoyment ... [and] accidental beatitude will receive some increase from the union of soul and body, and from the sight of the new heaven and the earth. cf. [ Heaven | New Advent].
Results of the General Judgment | New Advent:



  With the fulfilment of the sentence pronounced in the last judgment
  the relations and the dealings of the Creator with the creature find
  their culmination, are explained and justified. The Divine purpose
  being accomplished, the human race will, as a consequence, attain its
  final destiny. The reign of Christ over mankind will be the sequel of
  the General Judgment.



Against Heresies (St. Irenaeus) > Book V, Chapter 36 | New Advent:


Men shall be actually raised.
The world shall not be annihilated; but there shall be various mansions for the saints, according to the rank allotted to each individual. 
All things shall be subject to God the Father, and so shall He be all in all.
But when this present fashion of things passes away, and man has been renewed, and flourishes in an incorruptible state, so as to preclude the possibility of becoming old, then there shall be the new heaven and the new earth, in which the new man shall remain continually, always holding fresh converse with God.
Then those who are deemed worthy of an abode in heaven shall go there, others shall enjoy the delights of paradise, and others shall possess the splendour of the city; for everywhere the Saviour shall be seen according as they who see Him shall be worthy.
The just will be resurrected and inherit the kingdom of the earth.
The creation shall be free from the bondage of corruption, so as to pass into the liberty of the sons of God.
And God the Father bestowing in a paternal manner those things which neither the eye has seen, nor the ear has heard, nor has thought concerning them arisen within the heart of man.





Please see also:


Section VI. in the Catechism of the Catholic Church The Hope of the New Heaven and the New Earth, CCC 1042-1050.
Pastoral Constitution on the Church in the Modern World Gaudium Et Spes, 39, Promulgated By His Holiness, Pope Paul VI On December 7, 1965:



  
  We do not know the time for the consummation of the earth and of humanity,1 nor do we know how all things will be
  transformed. As deformed by sin, the shape of this world will pass
  away;2 but we are taught that God is preparing a new
  dwelling place and a new earth where justice will abide,3
  and whose blessedness will answer and surpass all the longings for
  peace which spring up in the human heart.4 Then, with death
  overcome, the sons of God will be raised up in Christ, and what was
  sown in weakness and corruption will be invested with
  incorruptibility.5 Enduring with charity and its
  fruits,6 all that creation7 which God made on
  man's account will be unchained from the bondage of vanity.
  
  
  1. Cf. Acts 1:7.
  2. Cf. 1 Cor. 7:31; St. Irenaeus, Adversus haereses, V, 36, PG,
  VIII, 1221.
  3. Cf. 2 Cor. 5:2; 2 Pet. 3:13.
  4. Cf. 1 Cor. 2:9; Apoc. 21:4-5.
  5. Cf. 1 Cor. 15:42 and 53.
  6. Cf. 1 Cor. 13:8; 3:14.
  7. Cf. Rom. 8:19-21.