Recently I got an offer letter for a master program (with courses like algebraic topology, algebraic geometry (first chapter of Hartshorne), differential geometry, representation theory, etc... ).
Now I have three months before I go to there and start my program. My question is, should I use these three months to review the fundamental subjects in mathematics, i.e Abstract Algebra, Topology, Analysis, and Linear Algebra or should I pre-study those master programs before I go?
To be honest, I am not firm enough in those fundamental subjects as I didn't do exercises at all when I took those courses (but I managed to score well, mainly because my school only gives questions in the lecture notes). Also, I have almost forgotten some results in linear algebra (like the change of basis etc.)
At first, I would prefer to revise (by doing exercise) first before going to some advanced courses. But then a friend of mine suggested me to study further courses, where he says if I am stuck somewhere, then I go back for related topics. (For example, if I need a change of basis in the study of manifolds, then I go back to revise it.)
Which one would be better for my case, given the time constraint and the mathematical background?
As I said in the comments, I think the important thing is that if you feel you need to go over basic material, you should do it at a higher level of sophistication than an undergraduate would.
If you had more time, you could do this in books that go into greater depth in algebra and real and complex analysis, but my recommendations here will focus on books that will allow you to move faster while also reviewing the undergraduate material you'd like to know better.
I would recommend you see if these books work for you:
They are meant to introduce you to basic graduate-level material in algebra and analysis, but start by reviewing more elementary subjects, such as linear algebra and metric spaces. (There are also sequels, Advanced Algebra and Advanced Analysis.)
I'd like to mention that because linear algebra in vector spaces is treated before modules in Basic Algebra, important facts about linear algebra (for example giving a clearer account of Jordan canonical form) are delayed until Chapter 8, and some of it is done in the exercises there. This material could have been expanded.
As supplements to those books by Knapp, I would suggest the following:
Real and Functional Analysis by Serge Lang. This covers topology, integration theory, functional analysis, differential calculus in Banach spaces (with the basic theorems on differential equations) and, quite briefly, manifolds. This could actually be an alternative to Basic Analysis.
Algebra, also by Serge Lang. I am suggesting this as a reference, not as a textbook to read. It is very well-organized for reading chapters independently of each other. Here you will find fuller explanations, for example, of the linear algebra material I mentioned above.
Real and Complex Analysis, by Walter Rudin. I would recommend Chapter 10 as a supplement to (or substitute for) Knapp's appendix on complex analysis. It can be read independently of the other chapters and covers very succinctly the most important material that would appear in a basic undergraduate course. (The rest of the book is also excellent, but I'm not actually recommending you read it now due to your time constraints.)
Added. Since you mention change of basis, let me explain how I remember this. If $e = (e_i)$ and $e' = (e'_i)$ are bases for the same space $V$, then the transition matrix from $e$ to $e'$ is the matrix $P$ whose columns are the vectors of $e'$ written in the basis $e$.
If $x$ is a vector, write $X$ for its coordinates in the basis $e$ and $X'$ for its coordinates in the basis $e'$. Now there is only one formula to remember: $X = PX'$. (Proof: $P$ is the matrix of $\operatorname{Id}_V \colon (V, e') \to (V, e)$.) If you remember this, you will find all the other formulas.
For example, say $T \colon V \to W$ has matrix $M$ in bases $e$ and $f$. What is the matrix $M'$ of $T$ in bases $e'$ and $f'$? Answer: We have $X = PX'$, $Y = QY'$ and $Y = MX$. What we want is $Y' = M'X'$. A simple calculation shows that we must take $M' = Q^{-1}MP$.
Likewise, say a bilinear form has matrix $M$ in basis $e$. What is its matrix $M'$ in basis $e'$? Answer: We have $X = PX'$, $Y = PY'$ and the bilinear form is given by $X^t MY$. What we want is $(X')^t M' Y' = X^t MY$. It follows that we must take $M' = P^t M P$.