It might be immodest of me to mention my own work, but there is quite a lot known about the pseudo-holomorphic curves in $S^6$. For example, it is known that the pseudoholomorphic rational curves are all algebraic and have area a multiple of $4\pi$ (and, yes, there are many of them). (It's true that the moduli space is not compact; however, because the area is quantized, the rational curves of a given area, can, in fact, be compactified to yield a compact moduli space, even an algebraic one.) As another example, every compact Riemann surface occurs as a pseudoholomorphic curve (possibly ramified) in $S^6$. (A student of Kevin Corlette, some years ago, proved that you can actually get every Riemann surface as an unramified pseudoholomorphic curve in $S^6$.)
For example, see my paper Submanifolds and special structures on the octonians, Journal of Differential Geometry 17 (1982), 184–232. (I regret very much the misspelling 'octonians' that permeates that paper.)
Added references: (January 2020) Recently, I was asked some questions about pseudo-holomorphic curves in $S^6$, and that inspired me to look at some of the literature that has been generated in the nearly 40 years since I wrote the paper above. A fair amount has been worked out, and anyone who wants to know more about the subject would benefit from consulting a few of the following papers:
J. Bolton, L. Vrancken, and L. Woodward, On almost complex curves in the nearly Kähler $6$-sphere, Q. J. Math. Oxford (2) 45 (1994), 407–427.
H. Hashimoto, Deformations of super-minimal $J$-holomorphic curves of a $6$-dimensional sphere, Tokyo J. Math. 27 (2004), 285–298.
J. K. Martins, Superminimal surfaces in the $6$-sphere, Bull. Braz. Math. Soc. (N.S.) 44 (2013), 25–48.
L. Fernández, The space of almost complex $2$-spheres in the $6$-sphere,
Trans. Amer. Math. Soc. 367 (2015), 2437–2458.
M. Dajczer and T. Vlachos, A representation for pseudoholomorphic surfaces in spheres, Proc. Amer. Math. Soc. 144 (2016), 3105–3113.
This is by no means a complete list of relevant articles, but these do cover a range of interesting developments of the results in my 1982 JDG paper.