First of all, there are two general types of free energy: Gibbs and Helmholtz. Helmholtz free energy is $F=U-TS$, whereas the Gibbs free energy has an additional term that accounts for the work done on the environment by the system, $G=F+pV$. If you're modelling a system in a vacuum or a gas, say, then you'd typically use Helmholtz, whereas in a solution you'd use Gibbs.
Now imagine you have an infinite number of atoms, all separated by an infinite distance from each other. You then bring them all together to form an infinitely large crystal (i.e. there are no surfaces). The change in internal energy $U$ per atom (or per molecule) is the cohesive energy, while the change in the free energy (typically per unit volume or per unit cell) is the bulk free energy.
Binding energy depends on the context. In the case of a bulk material, the binding energy is just the negative of the cohesive energy, i.e. it's the energy required to disassemble the crystal.