Short answers, pulled from the story.
Gautama Buddha's early teachers were Alara Kalama and Uddaka Ramaputta. They taught him formless jhanas or mental absorptions that involved suppressing activity to achieve a state where pleasure and pain waned completely.
Anapanasati is mindfulness of inhalation and exhalation as part of paying attention to one's body in quietude. It recommends cultivating the Seven Factors of Enlightenment including sati, dhamma vicaya, viriya, piti, passaddhi, samadhi, and upekkha.
The 5th-century Visuddhimagga was written by Buddhaghosa. This text became the most influential presentation of Theravada meditation theory despite containing non-canonical instructions.
Deity Yoga involves recitation of mantras prayers and visualization of yidam deity along with associated mandala. Practitioners imagine themselves becoming that body developing divine pride understanding that self and deity are not separate.
Psychologists and expert Buddhist meditation teachers including Dipa Ma Anagarika Munindra Thich Nhat Hanh Pema Chodron Clive Sherlock Mother Sayamagyi S.N. Goenka Jon Kabat-Zinn Jack Kornfield Joseph Goldstein Tara Brach Alan Clements and Sharon Salzberg have integrated these practices.