The process by which a Guru initiates a disciple into a mantra is called Mantra Diksha.
Is it mentioned in the Vedas about this process or about other forms of Dikshas?
No there is only one thing which is mentioned in Vedas (here in Atharva Veda)
Leaving humanity behind, making the heavenly word thy choice, With all thy friends address thyself to furthering and guiding men.
The Ganapati Atharva Seersha which is a part of Atharva Veda clearly mentions the need of initiation .
Quoting the needed portions from here:
Idham adharvaseersha masishyaya na dheyam, Yo yadhi mohad dhasyathi sa papeeyan bhavathi, Sahasravarthanath yam yam kamam adheethe, Tham thamanena sadhayeth.
Meaning-This adharva seersha should not be given, who is not a disciple, He who gives it due to passion, becomes a great sinner, Thousand times repetition helps one get all that one wishes to get.