Typically if you want to provide translations for a picklist, you can use the translation workbench or deploy an object translation configuration file through the metadata api. These translations will be displayed to the user in the UI, but the value will not be stored as language specific text.
When you are on the object setup page and you see the list of fields, when you click on the name of the picklist field (not on edit) you will see the list of api values for the picklist. Even if these values are not displayed to the users, these values will be what you get when pulling data using the api, no matter what language the picklist value was selected in.
As for how to enter translations, you could go to:
Setup > Translation Workbench > Translate