Thought Process behind Parametrics in the $(x,y,z)$ space Converting parametrics to a rectangular equation in $2$D is pretty straight-forward, I think: just solve for $t$ and set them equal to each other or do a substitution. 
$3D$ is confusing me, however. 
For example, $r(t) = (t, t, t^2)$ or $r(t) = (t, \sin t, 2\cos t)$
What steps would I take to visualize these as well as make the mathematical connection to the cartesian plane? Of course plotting points is possible, but tedious in $3$D. 
Others in my class are able to simply look at these and know what shape they make, something I do not know how to do.
 A: By all means, keep the parametrization!
Note that a curve in ${\mathbb R}^3$ has codimension $2$, hence is given by $2$ equations. As an example take the circle resulting from the intersection of the plane $x+y+z=1$ with the sphere $x^2+y^2+z^2=1$.
Instead try to visualize the given curve by following the points ${\bf r}(t)$ with your inner eye in real time while they are "produced". Take your
$$\gamma:\quad t\mapsto{\bf r}(t):=(t,\sin t,2\cos t)\qquad(-\infty<t<\infty)\tag{1}$$
as an example. Projecting to the $(y,z)$-plane we obtain an everlasting elliptical movement
$$t\mapsto\hat{\bf r}(t):=(0,\sin t,2\cos t)$$
with semiaxis $1$ in $y$-direction and $2$ in $z$-direction. According to $(1)$ the actual point ${\bf r}(t)\in\gamma$ in addition moves with constant speed $1$ in $x$-direction. It follows that the curve $\gamma$ is an "elliptical helix" with pitch $2\pi$.
A: Well, as throughout maths, it is a matter of experience ("trained eye") + genius.  
Concerning your examples:
$$
r(t,t,t^2 )\quad  \Rightarrow \quad \left\{ \begin{gathered}
  x = y \hfill \\
  z = x^2  = \frac{1}
{2}\left( {\sqrt {x^2  + y^2 } } \right)^2  = \frac{1}
{2}s^2  \hfill \\ 
\end{gathered}  \right.
$$
a parabola in the plane $x=y$  
and
$$
r(t,\sin t,2\cos t)\quad  \Rightarrow \quad \left\{ \begin{gathered}
  \frac{{x^2 }}
{{1^2 }} + \frac{{y^2 }}
{{2^2 }} = 1 \hfill \\
  z = \arctan \left( {\frac{x}
{{y/2}}} \right) + 2k\pi  \hfill \\ 
\end{gathered}  \right.
$$
an elliptical helix as already indicated by Christian.
So, carefully observe  the parametric eq., note the domain of definition / variability(!), extract known/simple relations between $x,y,z$ induced by the parametric, add the necessary conditions to respect the definition domain (in case splitting or trunking the cartesian eq.).
