First book on algebraic topology
Warning : the following books are not algebraic topology books in the classical sens, but anyway it's impossible to get in AT without any background in topology : these two books will give you the necessary background and go after in the beginning of algebraic topology.
"Introduction to Topological Manifolds" by Lee is very readable and starts from scratch. It covers classic topology (4 chapters), one chapter on CW complex and classic topics in algebraic topology (fundamental group, covering space and groups) with emphasis on surfaces (with classification of compacts surface) with finally last chapters about homology and cohomology.
Another reference is the book of Munkres, Topology, which covers with lot (and lot) of details general topology, and also fundamental group and covering space.
For both, no background is needed (There is an appendix in Lee about group theory, and a chapter consacred to Set theory in Munkres !). For more advanced topics in algebraic topology (homology theory for example, or differentiable topology) I think calculus in $\mathbb R^n$, general topology and abstract algebra are highly recommended.
The canonical reference is probably Hatcher's "Algebraic Topology," which in addition to being a very well-written text also has the advantage of being available online for free in its entirety. It stays in the category of CW-complexes for the most part, and there's a self-contained appendix describing enough of its topology to get you through the book. The only drawback I'd mentioned of it is that the more advanced topics (e.g., obstruction theory) covered in the appendices to its individual chapters often assume a bit more background or general sophistication than the rest of the book. It's not really a problem, and it's definitely not a reason to decide against the book, but you will probably notice a sharp difficulty spike if you go through those sections in order.
On the other hand, there's unfortunately no way of getting around the prerequisites in point-set topology. You'll find it easier after the first half of Hatcher, when things get more algebraic, but you should at least be comfortable with basic concepts like connectedness, compactness, metric spaces, various extensions lifting properties, etc. For that, I think Munkres' Topology is the canonical reference, but I can't recommend it to anyone; it's pedantic, tedious, and dull. I can't remember exactly what Lee's Introduction to Topological Manifolds, which N.H. mentioned above, covers in basic topology; still, it's a good book, and I'd recommend reading it regardless of whether you plan on continuing in algebraic topology.