How to calculate interpolating splines in 3D space?

I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a subset of other types of curves such as Bezier curve or the Hermite curve?

I have successfully found cubic splines in 2 dimensions, but I'm not sure how to extend it into 3 dimensions and why there is no explanation about this.

Is there a better and more documented type of curve I could use to achieve this? My goal is to move an object along the smooth curve going through the control points. Please help.

First, let's understand parametric splines.

Let's assume we already know how to find $y$ as a (spline) function of $x$. And suppose we have a sequence of 2D points $P_i = (x_i,y_i)$. First, we assign a parameter value $t_i$ to each point $P_i$. The usual way to do this is to use chord-lengths -- you choose the $t_i$ values such that $t_i - t_{i-1} = \|P_i - P_{i-1}\|$. Then you compute $x$ as a function of $t$. The calculation is the one you already know, but it's just $x=f(t)$ instead of $y=f(x)$. Now do the same thing with $y$ and $t$. So, now you have both $x$ and $y$ as functions of $t$. In other words, you have a 2D point $(x,y)$ as function of $t$, which means you have a 2D parametric curve.

Now, the extension to 3D is straightforward. We just make $z$ a function of $t$, also. So, now we have $(x,y,z)$ as a function of $t$, so we have a parametric space curve.

To do 3D spline interpolation using Matlab functions, see here.

A better reference is this web site.

Bezier curves are also easy to extend to 3D. As you probably know, the equation of a cubic Bezier curve is $$ \mathbf{C}(t) = (1-t)^3\mathbf{P}_0 + 3t(1-t)^2\mathbf{P}_1 + 3t^2(1-t)\mathbf{P}_2 + t^3\mathbf{P}_3 $$ In this equation it doesn't matter whether the control points $\mathbf{P}_0$, $\mathbf{P}_1$, $\mathbf{P}_2$, $\mathbf{P}_3$ are 2D or 3D.