Redox reactions consist of oxidation and reduction processes occurring simultaneously. What does this actually mean?
Redox reaction involves the transfer of electrons between species (Easy way to remember: increase in oxidation number = oxidation, hence decrease in oxidation number = reduction). These processes occur together at the same time at different interfaces (boundaries).
Considering your cell (Galvanic Cell): The main reacting species (excluding the spectator ions) are Copper and Zinc. The overall reaction that is taking place is:
$$ \ce {Zn + Cu^{2+} \rightarrow Zn^{2+} + Cu}$$
Applying the previously mentioned trick, you can see that zinc is being oxidised from 0 oxidation number to 2+ (increase in oxidation number = oxidised), whereas copper ion is reduced from 2+ to 0 oxidation number (decrease in oxidation number = reduced). You can think of a half-cell equation as small components that make the overall reaction in redox reaction of Galvanic cell. They provide vital information on what is happening to the species in terms of electrons involved and the standard potential can be used to identify if the species is capable of operating as reducing agent or oxidising agent.
The half-cell equations is usually tabulated against standard hydrogen electrode as being the reference.(Note: half-cell equations are usually written as reductions).
In order to identify which species is oxidising agent, you can think of it this way: the more positive the potential $\Rightarrow$ stronger oxidising agent $\Rightarrow$ oxidising agents by definition are reduced (reduction occurs at the cathode).
The reactions are only taking place when the circuit is closed (cells are connected). This leads to the flow of the electrons around the circuit and redox reaction can take place.
To qualitatively prove that your reaction occurs, you have to consider the cell potential:
$$E_{cell} = E_{cathode} -E_{anode} $$
applying the concept of electrical work = Gibbs energy, thus
$$G=-nFE_{cell}$$
(Important: provided the concentration 1 M, if they are not you have to apply Nernst Equation to identify new cell potential) .