Mainly cell type refers to the differentiation degree to form a cell population whose characteristics are identifiable by using simple microscopy, stains or immunological methods. Examples of cell types would be T8 lymphocytes, peripheral astrocytes or hepatocytes. Cell types can be pathological too, like tumor cells.
Cell state makes references to the current physiological condition of a given cell. A given cell type can be found in different proliferation states, metabolic conditions (i.e: An adipocyte or a hepatocyte will have different metabolic profiles depending on the insulin and glucagon balance), electrophysiological state (The human oocyte rapidly changes its membrane potential once a spermatozoid has entered) or even degree of differentiation (The typical example are B lymphocytes, which can be in a "naïve" state if they haven't found an antigen, or into a mature state. Once they have matured, thay can be subdivided into cells that actively produce antibodies and cells that divide actively).
A cell function usually refers to a process that occurs inside a cell, or to the physiological function that cell type has in the whole organism. Cell functions could be metabolism, signaling, cell division, etc.; while also could be the likes of detoxification (hepatocytes), hormone production (Langerhans cells) or contraction (myocytes).