By removing a card from your hand from the game. Does the card go in the trash pile or do we remove it from the game?
No, every card that you would put in the waste pule says for it to be placed in the waste pile.
A good example of this is resolving a crisis as you don't add those cards to the waste pile when you are done. Here is the online rules
Resolve Crisis: Shuffle the cards that were added face-down to the crisis during the player turns phase. Reveal them one at a time. Each item card added that has a symbol matching a symbol in the prevent section of the crisis is worth 1 point. Each card that doesn’t have a symbol matching a symbol in the prevent section of the crisis subtracts 1 point. After revealing all of the cards, if the combined point total is lower than the number of players then immediately resolve the crisis. If the point total equals or exceeds the number of players the crisis is prevented. Additionally, if the point total exceeds the number of players by 2 or more the colony gains 1 morale. After resolving the crisis, remove all cards added to the crisis from the game. See Example: Resolving a Crisis to lef