While the majority of Hindus believe in reincarnation, the view is not universally held among all Hindus.
The Bhagavad Gita certainly supports the belief in an imperishable soul. Here some of the key verses:
Chapter 2, verse 13
dehino'smin yatha deha kaumaram yauvanam jara tatha dehantara praptir dhiras tatra na muhyati
Translation: Just as in the physical body of the embodied being is the process of childhood, youth, old age; similarly in the transmigration from one body to another the wise are never deluded.
Chapter 2, verse 17
avinasi tu tad viddhi yena sarvam idam tatam vinasam avyayasasya na kascit kartum arhati
Translation: But know that by whom the entire physical body is pervaded is indestructible. No one is able to cause the destruction of the imperishable soul.
Chapter 2, verse 20
na jayate mriyate va kadacin nayam bhutva bhavita va na bhuyah ajo nityah sasvato yam purano na hanyate hanyamane
Translation: The soul never takes birth and never dies at any time nor does it come into being again when the body is created. The soul is birthless, eternal, imperishable and timeless and is never terminated when the body is terminated.
Chapter 2, verse 22
vasamsi jirnani yatha vihaya navani grhnati naro'parani tatha sarirani vihaya jirnany anyani samyati navani dehi
Translation: As a person gives up old and worn out garments and accepts new apparel, similarly the embodied soul giving up old and worn out bodies verily accepts new bodies.
Chapter 2, verse 24
acchedyo'yam adahyo'yam akledya'sosya eva ca nityah sarva-gatah sthanur acalo'yam sanatanah
Translation: The soul is indestructible, the soul is incombustible, insoluble and unwitherable. The soul is eternal, all pervasive, unmodifiable, immovable and primordial.
Chapter 2, verse 27
jatasya hi dhruvo mrtyur dhruvam janma mrtasya ca tasmad apariharye'rthe na tvam socitum arhasi
Translation: For one who has taken birth, death is certain and for one who has died, birth is certain. Therefore in an inevitable situation understanding should prevail.
I would view these verses as supporting the eternal nature of the soul. Many Hindus would also view these verses as supporting the literal transmigration of the soul from one physical body to another (ie reincarnation). Reincarnation or rebirth could also mean being reborn into a new life after physical death that is very different from the physical existence we once had.