In the first law of thermodynamics, we learned that W and Q are path-dependent quantities, but how are Q and W defined?

The first law of thermodynamics, is as the name suggest a law. It states that if you consider some process a thermodynamic system undergoes then
$\mathrm{\Delta <!--\; \Delta \; -->}U=W+Q$
The point is that W and U are things that depend on what you are studying. Pick the infinitesimal version just for simplicity
$dU=\mathrm{\delta <!--\; \delta \; -->}W+\mathrm{\delta <!--\; \delta \; -->}Q$
Then for a gas it trully makes sense to put $\mathrm{\delta <!--\; \delta \; -->}W=-<!--\; -\; -->PdV$, if you reason a little bit you will se this agrees with your intuition of mechanical work for a gas. For a rubber band under some tension $\mathrm{\tau <!--\; \tau \; -->}$ it makes sense to put $\mathrm{\delta <!--\; \delta \; -->}W=\mathrm{\tau <!--\; \tau \; -->}dL$. And there is not just mechanical work, but also chemical work.
The law ends up being something like: if you know for a certain thermodynamic system how to define it's energy and define work properly then heat will be the change in the energy which is not in the form of work.
Why to think in this way? Thermodynamics is concerned with the description of equilibrium states of macroscopic systems. These equilibrium states are defined as those which can be described with just some extensive parameters, without counting all degrees of freedom the system really has.
Inside W you count the change in the energy associated with changes on the measured degrees of freedom, the extensive parameters. But there is also change in energy due to changes on the unmeasured parameters. This change in energy due to tons of parameters together which you cannot control directly is then called heat and the first law states exactly that: the change in energy of a thermodynamic system consists of work and heat where work is change in energy due to change on the measured degrees of freedom and heat is the change in energy due to change on the unmeasured degrees of freedom.
Now, W and Q are not functions of state because one can see intuitively they are properties of a process and not of the states themselves.