Do Latter-day Saints believe that they will “get their own planet”?

No. This idea is not taught in Latter-day Saint scripture, nor is it a doctrine of the Church. This misunderstanding stems from speculative comments unreflective of scriptural doctrine. Latter-day Saints believe that we are all sons and daughters of God and that all of us have the potential to grow during and after this life to become like our Heavenly Father (see Romans 8:16-17). The Church does not and has never purported to fully understand the specifics of Christ’s statement that “in my Father’s house are many mansions” (John 14:2).

Frequently Asked Questions, Question 12: https://newsroom.churchofjesuschrist.org/article/frequently-asked-questions & https://www.churchofjesuschrist.org/study/liahona/2010/10/commonly-asked-questions?lang=eng

To live in the highest part of the celestial kingdom is called exaltation* or eternal life. To be able to live in this part of the celestial kingdom, people must have been married in the temple and must have kept the sacred promises they made in the temple. They will receive everything our Father in Heaven has and will become like Him. They will even be able to have spirit children and make new worlds for them to live on, and do all the things our Father in Heaven has done

Gospel Fundamentals, Chapter 36 “Eternal Life”, page 201: (see pdf version) https://www.churchofjesuschrist.org/bc/content/shared/content/english/pdf/language-materials/31129_eng.pdf