The Bhagavad-Gita, which is the essence of all vedic literature and knowledge and the direct word of god, Lord Shri Krishna, states a vast ammount of evidence on the fact of reincarnation. Krishna says that just as the body changes clothes from day to day, so too, does the soul change bodies from lifetime to lifetime. Just as the body changes from childhood, to teenager, to old age, so does our consciousness and soul change throughout various incarnations. Look at how different you are now, then how you were from your childhood, Youve got an entire new body. Your whole body has dissolved and reconstructed itself, changed its shape and is still going through change. We are always in a constant state of change. This is also verified by Buddha, not only Krishna.
Krishna Is the supreme lord, cause of all causes, the original source of all creation and destruction, inexhaustible. He states in the bhagavad gita:
"That which pervades the entire body is indestructible. No one is able to destroy the imperishable soul.
Only the material body of the indestructible, immeasurable and eternal living entity is subject to destruction; therefore, fight, O descendant of Bharata.
He who thinks that the living entity is the slayer, or that the entity is slain, does not understand. One who is knowledge knows that the self slays not nor is slain.
For the soul there is never birth, nor death. Nor, having once been, does he ever cease to be. He is unborn, eternal, ever existing, undying and primeval. He is not slain when the body is slain.
O partha, how can a person who knows that the soul is indestructible, unborn, eternal and immutable kill anyone or cause anyone to kill?
As a person puts on new garments, giving up the old ones, similarly, the soul accepts new bodies, giving up the old and useless ones.
The soul can never be cut into pieces by any weapon, nor can he be burned by fire, nor moistened by water, nor withered by the wind.
This individual soul is unbreakable and insoluble, and can be neither burned nor dried. He is ever lasting all pervading, unchangeable, immovable and eternally the same.
It is said that the soul is invisible, inconceivable, immutable and unchangeable. Knowing this, you should not grieve for the body.
If, however, you think the soul is perpetually born and always dies, still you have no reason to lament, O mighty-armed.
For one who has taken his birth, death is certain; and for one who is dead, birth is certain. Therefore, in the unavoidable discharge of your duty, you should not lament.
All created beings are unmanifest in their beginnings, manifest in their interim state, and unmanifest again when they are annihilated. So what need is there for lamentation?
(bhagavad gita, ch. 2, lines 17-28 )