The standard texts (Hatcher, May, etc.) cover material which was, in large part, understood by 1950, though this material is filtered - conspicuously so, in May's text - through the authors' modern perspectives and sensibilities. That leaves another half-century of development. So, as a follow-up to first-year algebraic topology - still far from the cutting edge, but very relevant to reaching it - may I recommend reading some of the classic papers of the mid-twentieth century?

Many are easy to find online. Several - no, many! - are written by great mathematicians and great expositors. For me, the material is the more exciting in the words of its discoverers. Many people will have their own favourites; my list is slanted towards differential topology.

A couple by Serre. Homologie singulière des éspaces fibrés has as clear and economical an account of spectral sequences as I've seen anywhere. The method of Groupes d'homotopie et classes de groupes abéliens might be considered old-fashioned, but it gives a strong taste of what localisation can achieve (e.g. where is the first $p$-torsion in the stable stems?).

Three papers that achieve perfect marriages of algebraic topology and differential geometry: Thom's Quelques propriétés des variétés différentiables founded cobordism theory. Kervaire-Milnor's Groups of homotopy spheres I essentially began surgery theory. Both are astonishingly far-seeing. Finally, Deligne-Griffiths-Morgan-Sullivan's Real homotopy theory of Kähler manifolds: minimal models are things you can build at home!