I have been told to study the mapping class group of the torus for a summer project from Dale Rolfsen's Knots and Links (Chapter 2-D) in preparation of studying 3-manifolds. However Rolfsen proof of the main theorem of the section, that $Aut(T^2)$ modulo ambient isotopy is isomorphic to $GL(2,\mathbb Z)$ has lot of gaps to be filled in as exercises. I lack the familiarity required with the material to be able to do this in a timely manner on my own. Hence I am looking for resources (books, lecture notes, video lectures etc.) that explain this theorem in detail and would help me in filling the gaps in Rolfsen's proof.
My advisor told me that we're not looking to study knot theory, we just need these tools to study 3-manifolds so I would also appreciate recommendations on books that cover similar content to Rolfsen's book but are more detailed.
As per OP's request, I am expanding my comment into an answer. It seems like there is a clear path to be taken: from surfaces to 3-manifolds, and possibly up to 4-manifolds. I will reproduce this here.
The Mapping Class Group of surfaces is generated by Dehn twists (Dehn's result, further improved by Lickorish on the minimal number of generators needed). For the case of the 2-torus, it is $\mathrm{SL}(2,{\bf Z})$. This is exposed nicely in $\color{brown}{\text{Clay and Margalit's }\textit{Office Hours with a Geometric Group Theorist}\text{, Lecture 17}}$. It has some pictures, and covers enough of the proof so that the gaps left are an actual easy exercise (for once!).
Regarding this, my guess is that your advisor has surgery theory in mind. Roughly-speaking, this is presenting 3-manifolds by means of links and numbers. Existence is the Lickorish--Wallace theorem, and uniqueness could be Kirby calculus in a sense. This is very well exposed in $\color{brown}{\text{Saveliev's }\textit{Lectures on Topology of 3-manifolds}}$. It reviews the theory in good details, and the proofs are quite clear and explanatory and accompanied by examples, special cases and pictures. What is great (and because you were asking about Knot theory in general too) is that Saveliev also talks about Knot-theoretic constuctions, namely: Seifert surfaces/matrices, the Alexander polynomial (with computations for torus knots), and fibered knots.
This theory is not limited, however, to 3-manifolds, and extends well to 4-manifolds. If you want to pursue in a Ph.D, this is the kind of stuff you will necessarily meet, assuming you do Geometric Topology in a broad sense. For this, $\color{brown}{\text{Gompf and Stipsicz's}\textit{ 4-Manifolds and Kirby Calculus}}$ will be your Holy Bible. This is a very difficult read, and proofs are so (con)dense(d) that they can eat up your time very fast. But it is a goldmine of examples and explicit descriptions of manifolds, fibrations, bundles, all with different points of vue. This will be for later, but you might want to give it a quick look if time permits!
As a side note, a very powerful tool that you will constantly use in Geometric Topology is the so-called Handle decompositions. There is plenty of information on the Wikipedia page I linked, but one thing to keep in mind if you know what a cellular decomposition is: they are just glorified CW-complexes made to work with manifolds (a handle decomposition even deformation retracts onto a cellular decomposition). They arise from Morse functions just the same, and they are the main bulding blocks that are used to talk about surgery presentations of 3-manifolds and 4-manifolds (a.k.a. Kirby diagrams).