This is a great question!
There is a big difference between the New Age version of reincarnation and that of Eastern religions such as Hinduism and Buddhism that teach this doctrine about life after death.
In the Eastern version, reincarnation is all about karma and the assumption that a person's actions have both positive and negative consequences in the next life and any future incarnations. The object is to live a good life and perfect oneself in order to achieve moksha, which is to be set free from the cycle of death and rebirth known as samsara. Hindus believe that the better one lives his/her life, the better the chance of achieving moksha. Indiscriminate living, on the other hand, leads to continued bondage to the cycle of death and rebirth and the chance of being reborn as a plant or animal.
Reincarnation to New Agers is entirely different, mostly because there is no concept of evil or immorality in the New Age - it's just a matter of making good or bad choices. So the object of reincarnation to a New Ager is to bring a person into a deeper understanding of who they are.
Whereas Buddhists and Hindus believe that being reborn as a human means they are making progress toward perfection, New Agers do not believe that a person can be reborn as a plant or animal. They are always reborn as a human.
Also unlike the Eastern version of reincarnation, the New Ager believes they have some control over how and when they are reincarnated.
Evidence to support the theory of reincarnation is nonexistent, unless you believe in the bogus science known as "past life regression therapy" where all kinds of false memories are planted in people by their therapists.
Aside from the lack of empirical evidence, the Bible is full of arguments against reincarnation.
For example, in the Old Testament, the book of Job tells us: " . . . (S)o men lie down and rise not again. Till the heavens are no more, they shall not awake, nor be roused out of their sleep." (Job 14:12)
In the New Testament, Jesus Himself tells us that there is no "second chance" at life here on earth, most notably in the story of the poor beggar named Lazarus who was continually ignored by a rich man while the two lived on earth. When both men died, Lazarus went to heaven but the rich man went to hell. The rich man later begged Lazarus for water, but was refused. "Abraham replied, "My child, remember that you received what was good during your lifetime while Lazarus likewise received what was bad; nut now he is comforted here, whereas you are tormented. Moreover, between us and you a great chasm is established to prevent anyone from crossing who might wish to go from our side to yours or from your side to ours." (Luke 16: 25-26).
This is why the Church teaches that there is no reincarnation. "Death is the end of man's earthly pilgrimage, of the time of grace and mercy which God offers him so as to work out his earthly life in keeping with the divine plan, and to decide his ultimate destiny. When 'the single course of our earthly life' is completed, we shall not return to other earthly lives: 'It is appointed for men to die once.' There is no 'reincarnation' after death" (CCC No. 1013).