I am trying to visualize lens spaces geometrically.

While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious limitations, some of them are not too complicated; For example, the 3-sphere can easily be seen from its simpler Heegaard splittings, or even directly from the compactification of $\mathbb{R}^3$. The Klein bottle is a bit more tricky, but can sort of be visualized if we think of the fourth dimension as a change in colour, for example.

However, when it comes to Lens spaces, I'm lost. I know several definitions for those spaces (to clarify, I mean the 3-dimensional case) - there's the "standard" definition which uses the formula for the 3-sphere embedded in $\mathbb{R}^4$ with a ${\mathbb{Z}}/{p\mathbb{Z}}$ action. Another definition is via Heegaard splittings, there's a definition which uses identification of triangles on a 3-ball (or pyramid). Finally, there's a definition using Dehn surgery.

The definitions can be found here: http://en.wikipedia.org/wiki/Lens_space

The first definition is completely non-intuitive for me (I can't even visualize $S^3$ as embedded in the complex plane). The Heegaard splitting definition is simple, but unclear geometrically as well. I think the one using identifications on a ball is the most promising, however when I'm trying to draw and visualize it I get something which doesn't feel like a manifold (and not for non-embedding reasons), so clearly I'm doing something wrong.

So could someone suggest a way to visualize those spaces? Pictures and/or references would also be of great help.