I think that it may be best to finish studying in Algebraic Topology and then start Differential Topology.
My experience is this: To gain a deep understanding of differential topology and the power of its abstract nature you need to have a good and working experience with 1 and 2 manifolds, namely curves and surfaces (my favourite book on the area is probably Tapp's Differential Geometry of Curves and Surfaces) which can be easily visualized and provide a great deal of motivation.
On the other hand I think that Algebraic Topology doesn't really have such a prerequisite (other than some familiriaty with General Topology which I undestand you have). Also, at least in some sense, knowing Algebraic Topology will help you with Differential Topology but not the other way around (one very good book to do so is Massey's : An introduction to Algebraic Toplogy). Nevertheless, you may be willing to learn about the "inbetween" area of differential forms and De-Rham cohomology (and for that I would suggest for first reading Bachmann's "A geometric approach to Differential Forms" and then a somewhat sterile but very complete book of Madsen & Tornehave: From Calculus to Cohomology)
Nevertheless, if you want to try flexing your muscles in Differential Topology Guillemin & Pollack offer a very readble introduction. For my though , Spivak's Differential Geometry Vol 1 did the trick and made me reallize the intrinsic beauty of the subject (and it also has a chapter dedicated to Algebraic topology too!)