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. <i>Homologie singuli&egrave;re des &eacute;spaces fibr&eacute;s</i> has as clear and economical an account of spectral sequences as I've seen anywhere. The method of <i>Groupes d'homotopie et classes de groupes ab&eacute;liens</i> 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 <i>Quelques propri&eacute;t&eacute;s des vari&eacute;t&eacute;s diff&eacute;rentiables</i> founded cobordism theory. Kervaire-Milnor's <i>Groups of homotopy spheres I</i> essentially began surgery theory. Both are astonishingly far-seeing. Finally, Deligne-Griffiths-Morgan-Sullivan's <i>Real homotopy theory of K&auml;hler manifolds</i>: minimal models are things you can build at home!