I need a function (a curve -- preferably a simple one) that, given $n$ points of a 2D space ($R^2$) passes (interpolates) through all points in a smooth/continuous way.
Found out that what I need is a spline, however cannot find one that behaves how I need. $n$-degree Bezier curves and B-splines don't pass through the $n$ points, just move from the first to the last one using the others as control points.
A Bezier spline of $n-1$ Bezier curves (defined between each pair of adjacent points) should do the trick, but to have a smooth result I would need some control points which I don't have and don't know how to compute.
Any hints?
A parametric cubic spline is what you want; I've talked about them at some length in this answer (and in probably a few other places on this site, since it seems to be a common question), so I won't repeat myself now. Suffice it to say that yes, Bézier isn't what you need here.