A photo of a Chinese typewriter, most of the disadvantages of Chinese script have been offset by technology these days, and there are a number of unexpected advantages (photo thanks to george morgan).
Being about to make a presentation to a number of people that includes my experiences of learning Chinese online, I thought I aught to address a couple of questions that are always asked to me.
Yes Chinese reading and writing does really involve learning thousands of distinct characters, unlike languages that have a phonetic alphabet. Assuming you are learning to read and write language that you already know this is not as bad as it sounds.
No Chinese in not hard to read and write on a computer, in fact is about as easy to write Chinese as English for many users via phonetic input systems (usually via pinyin). There are other input systems that are harder to learn (for example wubi). An expert in wubi probably inputs considerably faster then an English typer.
Displaying Chinese characters on your computer used to involve installing something but most systems these days have at least one Unicode font which means you can usually read Chinese and other non-Latin languages (that of course doesn't guarantee that you will understand it). If you are seeing little boxes then is means that you don't have a suitable font, if lots of question marks or garbage that probably means that the program you are using doesn't realize the character you are reading are in unicode or utf8 etc.
As far as webpages go then utf8 is almost certainly the best way to output your content.
Chinese dictionaries are hard to use, if you are looking for an unfamiliar character in a paper dictionary then ideally you need some knowledge of how Chinese characters are written. Characters have a series of common elements called radicals, if you can determine the radical and the number of additional strokes then you can look up the character, the more you understand how Chinese characters are constructed the easier it is to look them up (a bit of a catch 22 for the learner of Chinese). Computer dictionaries are somewhat easier you can cut and paste, look up characters by pinyin sound, and in some cases even draw an approximation. Many paper dictionaries will also allow you to look up a word by pinyin sound, but this only helps if you know what the word sounds like... The bottom-line for a Chinese learner (I many other languages I would guess) is that working with text on a computer for learning purposes is much faster, apart from browser plugins and on-line dictionaries, excellent software such as Stardict can smooth the way.
If you want to learn Chinese I would suggest starting learning to get used to the sounds first. I have also started to put together a collection of learning resources on in a twine, hopefully I will have some more specific advice soon but it will of course be from my own personal viewpoint.