Have a look at this page about Perlin Noise, in particular the "Interpolation" section. The general idea is that instead of a linear transfer function over t in [0, 1], you can apply something to result in smoother curves. The "smoothest" noise is a cos(t) function, but cubic or quintic polynomials can be used to approximate a cosine.
You're looking for Bezier Curves, or some other similar parametric curve. These are programatically quite easy to code and have the advantage of being intuitively straightforward to manipulate. The best treatise I know of is in the classic book Mathematical Elements of Computer Graphics, but any textbook on computer graphics will probably include a basic introduction.
What you're looking for is a Catmul-Rom spline, it's a type of Hermite spline that passes though the control points. Bezier curves are not the way to go, they are difficult to control in this situation.