  Let $X$ be a $d$-dimensional Gaussian process in $[0,1]$, where the component
are independent copies of a scalar Gaussian process $X_0$ on $[0,1]$ with a
given general variance function
$\gamma^2(r)=\operatorname{Var}\left(X_0(r)\right)$ and a canonical metric
$\delta(t,s):=(\mathbb{E}\left(X_0(t)-X_0(s)\right)^2)^{1/2}$ which is
commensurate with $\gamma(t-s)$. Under a weak regularity condition on $\gamma$,
referred to below as $\mathbf{(C_{0+})}$, which allows $\gamma$ to be far from
H\"older-continuous, we prove that for any Borel set $E\subset [0,1]$, the
Hausdorff dimension of the image $X(E)$ and of the graph $Gr_E(X)$ are constant
almost surely. Furthermore, we show that these constants can be explicitly
expressed in terms of $\dim_{\delta}(E)$ and $d$. However, when
$\mathbf{(C_{0+})}$ is not satisfied, the classical methods may yield different
upper and lower bounds for the underlying Hausdorff dimensions. This case is
illustrated via a class of highly irregular processes known as logBm. Even in
such cases, we employ a new method to establish that the Hausdorff dimensions
of $X(E)$ and $Gr_E(X)$ are almost surely constant. The method uses the
Karhunen-Lo\`eve expansion of $X$ to prove that these Hausdorff dimensions are
measurable with respect to the expansion's tail sigma-field. Under similarly
mild conditions on $\gamma$, we derive upper and lower bounds on the
probability that the process $X$ can reach the Borel set $F$ in $\mathbb{R}^d$
from the Borel set $E$ in $[0,1]$. These bounds are obtained by considering the
Hausdorff measure and the Bessel-Riesz capacity of $E\times F$ in an
appropriate metric $\rho_{\delta}$ on the product space, relative to
appropriate orders. Moreover, we demonstrate that the dimension $d$ plays a
critical role in determining whether $X\lvert_E$ hits $F$ or not.
