What do "S" and "F" markings on this transformer mean?
S for start and F for finish identifies the polarity of the windings.

For voltages like mains, there is no need to consider the phasing of S and F with respect to live and neutral, both sides of the winding will be insulated to a much higher voltage than mains anyway (this will not apply to kV transformers). 

If you use multiple windings on the transformer, then it will be important to pay attention to start and finish, from one winding to another winding.

With a split primary, you have two 120v windings, that you connect in parallel for 120v use, and in series for 240v. Here you must get S and F right. For 120v , connect both S's together, and both F's together. For 240v, connect an S to an F, and apply mains to the other S and F.

Much the same consideration goes to using a split secondary, and series or parallel connection.