Do I need to separate an object in order to give it different materials and textures? So I have a certain object that consists of multiple parts everyone of which has a different color/material. Do I need to separate those parts in editing mode or can I just add different textures to one object?
No you do not need to separate the object. 

Just go into edit mode and select the faces you want, then click the + button in Materials and add new material to selected faces by hitting assign.