Which translate and rotate mode should I be in when doing basic biped terrestrial animation;
Object, World , Gimbal ?
Yes all of them, whatever is most comfortable for you to get the pose you want.
Sometimes World is best for translating IKs, or center of gravity controls. Alternatively, with translate, Parent mode matches the graph editor keys more closely, while with rotate, Gimbal matches the graph editor. So when you want to visually animate a single channel (x, y, or z), then switching to Parent/Gimbal can be useful.
Most of the time Object mode is what you want, since it matches the object’s rotation. However it can be difficult to understand what channels you are animating exactly with Object mode since it depends on the parent transforms as well.
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That was HUGE. Thanx !
When you say parent mode, which one is that ?
In the tool settings with the translate/move tool active, under the axis orientation dropdown, the one that says “Parent” 
Thanks ! Im running Maya 2019 and don’t have the parent option under axis orientations.