The main under-armour clothing was a specialised arming jacket. These were usually made of multiple layers of cloth, and could include a layer of padding (cotton or silk wadding) - similar construction was used for European arming jackets (e.g., arming doublets). Some examples:
https://i.pinimg.com/originals/1b/57/06/1b5706511434cddf4d7ff47024554e8b.jpg
https://www.metmuseum.org/exhibitions/listings/2009/art-of-the-samurai/photo-gallery (see image 13)
https://i.pinimg.com/originals/cc/21/be/cc21be9a28f07531b68b03291869668c.jpg
Pants were also worn. These were short, reaching to about knee level. Examples:
https://commons.wikimedia.org/wiki/File:Samurai_kobakama.JPG
https://commons.wikimedia.org/wiki/File:Antique_Japanese_(samurai)_hakama.jpg
In addition, multilayered and/or padded short vests or shoulder pieces were worn, called manchira. These often included integral metal armour, to protect the neck and parts of the torso that would be otherwise exposed by gaps in the armpits and the tops of the shoulders. The armour could be mail, or small places sewn into the layers of cloth (like a Medieval European jack).
A vest type: https://www.metmuseum.org/art/collection/search/22133 (with an armoured neck)
Shoulder pad type: front: https://commons.wikimedia.org/wiki/File:Kikko_manchira.JPG and back: https://commons.wikimedia.org/wiki/File:Kikko_manchira_5.JPG
Shoulder pad with wrap-around pieces to cover the armpits: https://commons.wikimedia.org/wiki/File:Manju_no_wa_7.JPG
Arm armour was usually multilayer textile sleeves with integral metal armour, usually mail and/or plates covering the outer part of the arm, often extending to cover the back of the hands. These didn't need any further arming clothes under them (note that the arming jackets had sleeves).
Various supplementary armour could be worn besides or in addition to the manchira. For example,
armpit protectors: https://commons.wikimedia.org/wiki/File:Kikko_wakibiki_1.JPG
armoured pants: https://i.pinimg.com/originals/0b/cf/92/0bcf92c2a7775210a7b1fb719990f5f5.jpg
Finally, a nice animation by the V&A Museum showing the process of putting on armour, including the arming jacket and pants: