Can I study continuous time Fourier Transform and treat the rest as special cases
Yes you can, somehow. But that won't be the easiest road.

When teaching basic DSP, I often see students afraid to "learn all the formulae" and properties pertaining to the Fourier transform, Fourier series, discrete-time Fourier transform, $z$-transform and the Discrete Fourier transform with the fast
 Fourier transformation or FFT, up to time-frequency transformations. 

They share a lot of common traits: energy conservation, sorts of orthogonality, scalar product formulae, convolution turning into multiplication, Parseval-Plancherel formulae, etc. but also important (and often subtle) differences. 
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One tool to unite such concepts is sometimes called the Pontryagin duality. For a basic overview, the wiki page is ok:


  Pontryagin duality places in a unified context a number of
  observations about functions on the real line or on finite abelian
  groups


For instance, you can view the four primal and dual domains for Fourier "transformations" as in the table:

\begin{array}{lcc}
\textrm{Transform} &amp; \textrm{Original domain}&amp; \textrm{Transform domain}\\
\textrm{Fourier transform} &amp; \mathbb {R}  &amp;\mathbb {R} \\
\textrm{Fourier series}&amp; \mathbb {T} &amp; \mathbb {Z} \\
\textrm{Discrete-time Fourier transform (DTFT)} &amp; \mathbb {Z}  &amp; \mathbb {T} \\
\textrm{Discrete Fourier transform (DFT)}&amp; {\displaystyle \mathbb {Z} /(n)} &amp; {\displaystyle \mathbb {Z} /(n)}
\end{array}

Its provides several ways to derive properties from one domain to another.
But this tool dives deep into complicated concepts such as category theory, see for instance Notes 2: The Fourier transform (Terry Tao) or Category theory applied to Pontryagin duality.

Knowing properties about one transformation, you can derive properties of others, but for that you will have to master some advanced mathematics. There is no free lunch, unfortunately.

You would better learn how to remember a few formulae for a handful of transformations