Most important domains, extension theorems, and functions in several complex variables

Here are a few points to guide you into the beautiful subject you had the good taste to choose.

1) Hartogs extension phenomenon :given two concentric balls in $ mathbb C^n$, any holomorphic function $ B(0;M) setminus B(0;m) to mathbb C$ extends to a holomorphic function $ B(0;M) to mathbb C$.This really launched the subject and showed that function theory in several variables is not just an extension of the theory in one variable.You should study a few such classes of examples. Key words: Hartogs figures, Reinhardt domains.

2) Domains for which such extensions do not exist are called holomorphy domains: balls are holomorphy domains but as we just saw "shells" $ B(0;M) setminus B(0;m)$ are not.
If a region is not a domain of holomorphy, it has a holomorphic hull, but this is no longer included in $mathbb C^n$ : you get étalé spaces ( Yes, you algebraic geometers out there, this is where they were introduced ! ). This is an important subject and you can test whether a domain is a holomorphy domain at its boundary. Key-words: Levi problem, plurisubharmonic functions.

3)Holomorphic manifolds and Stein manifolds: these are abstractions of domains of $mathbb C^n$ and holomorphic domains respectively . In retrospect they were inevitable because of the nature of holomorphy hulls (cf. 2). Stein manifolds are highly analogous to the affine varieties of algebraic geometers.

4) Sheaf theory, cohomology: these are all powerful techniques that you MUST master if you want to read anything at all in the subject. In particular you must understand coherent sheaves, which have a flavour of Noetherianness in them, but are a more subtle notion.
The most important result here is Cartan's theorem B : coherent sheaves have no cohomology in positive dimension.

To help you learn all this , I would recommend:

B.Kaup, L. Kaup: Holomorphic Functions of several Variables (de Gruyter). [Quite friendly]

H.Grauert R.Remmert: Theory of Stein Spaces (Springer). [The ultimate source by the Masters]

All my wishes for success in your study of complex geometry.