Which quantities change when choosing a different standard state?
Quantities that are dependent on the standard state

When calculating the equilibrium constant K or the reaction quotient Q from a set of observed concentrations, these concentrations are given relative to the standard concentration of the species. For example, in aqueous solutions the solvent water has a different standard concentration (that of pure water) than solute (roughly 1 M). Therefore, the value of K and Q depend on choice of standard state.

Whenever a thermodynamic quantity has a Plimsoll symbol appended, it refers to a system at standard state, e.g. $\Delta_r G^\circ, \Delta_r H^\circ, S_f^\circ, \Delta H_f^\circ$ or to the substance at standard state. The choice of standard state changes the value of these quantities. For the biochemical standard state, the quantity is appended with a prime, e.g.  $\Delta_r G^\circ {'}$ to indicate that difference.

Quantities that have special values at standard state

At standard state, activities are equal to one. Consequently, the reaction quotient Q is also equal to one. 

$$Q = 1 \ \ \ \ \text{(at standard state)}$$Also, and this is a bit of a tautology but useful in deriving some relationships, when concentrations are at standard state (or more generally if $Q = 1$), quantities are equal to the standard quantities. Take for example the Gibbs energy of reaction:

$$\Delta_r G =  \Delta_r G^\circ + R T \ln Q$$

When concentrations are at standard state, this reduces to:

$$\Delta_r G =  \Delta_r G^\circ \ \ \ \ \text{(at standard state)}$$

Of course, that is the idea in calling $\Delta_r G^\circ$ the standard Gibbs energy of reaction.