Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear or nonlinear elliptic system. Further more, may I say that we can parallelly extened the $L^2$theory of the secondorder scalar elliptic equations to the corresponding $\mathbb{L}^2$theory of the secondorder elliptic systems? Any answer and reference will be appreciated!

All you mentioned can be extended to elliptic systems. The only things you should be careful about are the maximum principles and DeGiorgiNashMoser type regularity results. This means for linear systems the theory is very satisfactory. You can start with strongly elliptic systems and the variational approach to them, because the theory is much simpler and it covers a lot of important systems already. A good place to start would be McLean's book, Wloka's book, and perhaps Nirenberg's article. Wloka's book also contains a comprehensive introduction to the general elliptic theory, in the style of AgmonDouglisNirenberg. 

