Deriving thermodynamic quantities from partition functions

Let’s say I have a canonical partition function for the canonical assemble related to the Helmholtz free energy $A$, given by
$$A=-kTln Z$$

Now, I want to derive thermodynamical quantities, like the internal energy $E$, pressure $p$ and whichever thermodynamic quantity I want.

How do I go about this?

I know any thermodynamic quantity $X$ can be obtained by
$$langle X rangle = sum_{v} P_v X_v$$ where $v$ is an index of a permissible microstate.

For example, how would I get average energy $E$ or average pressure $p$ from such an equation?

So I know, from the above equation, I know
$$ Z = sum_{i} exp (-beta E_i – beta p_i V) implies P_i propto exp (-beta E_i – beta p_i V)$$

So, $$langle E rangle = sum _i P_i E_i = frac{-frac{dZ}{dbeta}}{Z}$$

I can do the same for pressure, but the differentiation can be done by $beta V$. How would I find say entropy $S$ for example?