Algebra Prerequisites: A knowledge of the following results:
(1) The definition of ring, subring, ideal and quotient ring.
(2) The correspondence theorem in ring theory.
(3) The notion of a prime ideal, of a maximal ideal, and the facts
that an ideal $I$ of a commutative ring $A$ is prime (resp. maximal)
if and only if $R/I$ is an integral domain (resp. field).
In short, the first 4 pages of Atiyah and Macdonald should be in the nature of a review for you.
(4) An extensive knowledge of field theory and Galois theory; for
example, in addition to the elements of Galois theory, you should
probably be familiar with separable and inseparable extensions and
transcendental extensions. (Chapter 5 on integral extensions of
commutative rings is better appreciated if you have already studied
the theory of algebraic extensions of fields. Transcendental
extensions are discussed in the chapter on dimension theory. Finally, at least one result in chapter 9 (on Dedekind domains) and a few exercises in chapter 5 require a knowledge of separable and inseparable extensions in field theory.)
(5) I think the Jordan-H\"older theorem in group theory is alluded to at some point in the text. (The discussion of modules of finite length in chapter 6.)
Topology Prerequisites: A knowledge of the following results:
(1) The definition of a topological space, of open and closed sets, of
a basis for a topology, of compact sets and Hausdorff spaces, of
subspaces, and of continuous functions. (In the text itself, point-set
topology is most prominent in the chapter on completions but you will
need point-set topology for the exercises as well.)
(2) Urysohn's lemma is needed to solve exercise 4 in chapter 4 and at
least one exercise in chapter 1 (on the characterization of the
maximal spectrum of a commutative ring).
Summary: The most important prerequisites are point-set topology and the theory of fields. You can read chapters 1-4 of Atiyah and Macdonald with only (1)-(3) of the Algebra Prerequisites above and chapter 5 of Atiyah and Macdonald with a knowledge of algebraic and separable extensions of fields. Chapter 9 of Atiyah and Macdonald also requires a knowledge of separable extensions of fields and chapter 10 of Atiyah and Macdonald requires a knowledge of (1) of the Topology Prerequisites above.
However, in order to do the exercises in Atiyah and Macdonald (which are the most important part of the text, in my opinion), you will need all the prerequisites above. Point-set topology is an essential prerequisite in the exercises because many exercises discuss affine schemes. Also, the elements of Galois theory are needed in some exercises in chapter 5, for example.
I hope this helps!