Another aspect of reading Chinese characters, which I've previously only mentioned in passing, is the issue of fonts and calligraphy. I'll say upfront that reading cursive/decorative Chinese calligraphy is difficult, and I cannot do it [see sidetrack in footnote]. So I'm going to stick to discussing fonts.
Chapter 12 of Douglas Hofstadter's excellent book Metamagical Themas has a couple of relevant figures (12.3 and 12.4, if you happen to own the book). The first of these shows the Latin letter A in various decorative fonts (the fonts.com website offers a similar set of examples), while the second does exactly the same for the Chinese character 黑 (hēi/black); see my photo of part of the page. The issue Hofstader is exploring with these figures is that of creating a font-making machine that can generate all possible versions of the letter A while also excluding everything that is not an A. However, the figures also serve to illustrate the fact that a reader who is very familiar with a particular set of graphemes (in this case, the Latin alphabet and Chinese characters, respectively) will have a much easier job separating out the decorative flourishes of a particular font from the underlying structural/meaningful parts.
In short: the more practice you get at reading Chinese characters in different fonts, the better you will be at it.
When I started learning to read Chinese menus, I got tripped up a lot by even very simple variations in the way a given character was depicted in different fonts. For example, the character 包 (bāo/package/bundle/bun) has a completely enclosed rectangular area in the centre in some fonts, yet in other fonts this area is open at the left-hand side (screenshot). It took me some time to properly convince myself that it was still the same character. Another one like this is 拌 (bàn/mixed), which in some fonts has the two strokes at the top on the right-hand side pointing inwards at the top, and in other fonts has them pointing outwards at the top (screenshot). Again, it took me some time to recognise these as the same thing.
I can mostly deal with these sorts of variation now, but every so often I still have to check whether a character really is the one I think it might be. I usually do this by pasting it into a Word document and viewing it in a couple of different fonts; the ones I mostly use are 儷宋 Pro and 华文楷体. I have no particular reason for choosing these, just that they happen to be installed on my Mac and they look fairly different from each other.
Relatedly, Chinese-Tools.com has a calligraphy editor that you can use to play around with viewing familiar characters in different fonts (note that the options in section 3 and the final option in section 1 will show you simplified characters rather than traditional ones). Some of these fonts are more like handwriting than printing, but it's still interesting to see the variations.
(Update, July 2013: see also Simon Cozens' post.)
Footnote: I also have more trouble than I should reading things handwritten in English, which is my native language. This is partly because I rarely read handwritten text any more, so I'm out of practice. My own handwriting (example photo) is not actually handwriting as such, since it's not cursive, but rather what we used to call "printing".