I keep reading about the stages of the cell cycle affecting stem cell fate, however I do not understand how. After all, any cell will have to pass through the whole of the cell cycle, and go through every state.
If you are referring to cells that are differentiated in adults: Some somatic cells in multi-cellular organisms are post-mitotic, that is, the cells cannot divide, or said in other words, they don't enter in S phase. This cells are said to be in G0 phase, because they keep synthesizing proteins but the won't replicate their DNA. Another thing we could say about cell cycle having something to do with cell division, is that in G1 and G2 (and , of course in G0) different cell types, produce different proteins. ie: a skeletal muscle cell, produces lots of proteins associated with the contraction, but neurons not so much.
If you are referring to the first cell divisions in a multi-cellular organism's development, then: in cleavage, the cells are dividing synchronously, and they avoid some cell cycle stages, like g1 or g2 (in cleavage, the cells are just dividing and replicating their DNA; the protein synthesis happens with mRNA inherited from the mother). In this stage, there is no cell growth at all. After some cell divisions, they have to start synthesize mRNA and, because of that, they start to divide asynchronously. Eventually the cells (for this and also for many other reasons) get differentiated.
I hope this answer is useful :)