Any PDE can have at least three types of nonhomogeneities: source, initial condition, and boundary conditions. Through linearity we can deal with each of them separately. The most important is the initial condition since it allows you to quickly write the Green's function.
In separation of variables you assume that the solution is a product of two functions that depend on each independent variable separately. Quickly this leads to a system of uncoupled ODEs from which one is a boundary value problem (please revise it from the ODE course).
The aforementioned boundary value problem will have countably many solutions satisfying the boundary values. This will also fix the separation constant.
The next step is to form an infinite superposition of all partial solutions and apply initial conditions (since a linear combination of solutions is a solution). This leads to Fourier series (please revise it). Note that you do not have to remember all the formulas for coefficients however, you must know how to derive them through orthogonality. This certainly is the best way.
Separation of variables works only for homogeneous initial conditions.
When dealing with nonhomogeneous source we use an ansatz of the exact solution. This is motivated by previous results from separation of variables. Please read the detail in the notes.
Nonhomogeneous boundary conditions are solved with a separation of the solution into two terms: $u = U + v$ from which $U$ satisfies the boundary conditions while $v$ is the remainder. The latter, will satisfy homogeneous BCs and possibly nonhomogeneous PDE with honhomogeneous IC. At that point we know how to solve them.
Using Green's functions helps you to write a solution to any type of linear problem. However, you should first calculate it which may be difficult.
Keep in mind that separation of variables is very robust, i.e. it can be utilized in any linear PDE with constant coefficients. Further examples are presented in the notes in sections concerning Poisson's and wave PDEs.
Remember that the Green's function is different for different problems, i.e. PDEs + IC + BCs. It is a common mistake that people use Green's function for the first problem that they learn (homogeneous Dirichlet BCs) and use it everywhere. Please do not do this.
Green's function for heat equation on the interval can be found by plugging in formulas for Fourier coefficients into the solution and interchanging order of summation and integration.
When solving Problem 13 it is very convenient to be able to solve Problem 12, i.e. to look for self-similar solutions of the form $u(x,t) = t^a U(x t^b)$ and then to choose $a$ and $b$ in order to satisfy PDE and boudnary conditions.