The article The TeX Family in 2009 appeared in the March 2009 issue of the American Mathematical Society's Notices. This page contains links to things referenced in that article, as well as some additional pointers that you may find useful.
Before reading on, please note that the best way to get TeX software is through a full distribution, a bundling of the materials for your computer; see below. If you have a recent distribution then it has a package manager, which supports download and installation with point and click. For example, both of the free distributions TeX Live and MiKTeX have package managers.
These things were discussed in the article (things mentioned only briefly are linked to below).
What TeX can do. One more extensive discussion of TeX's advantages is Why TeX?.
If you are a TeX beginner, then the starter page contains a short list of the things that you need to get running. In particular, the Not so short guide to LaTeX2e is a great tutorial. If you are an old hand at TeX but would like some recommendations about current standard tools, see this brochure.
Graphics. One way to specify your graphics inside of your TeX source file is PSTricks. It has extensive documentation, including a user guide containing many examples, and it has many contributed packages. Another way to specify your graphics is TikZ and PGF. There are many examples of this package in the extensive manual and this tutorial gives a gentle start. Another approach is to use the MetaPost language which has some unique and powerful capabilities for making technical graphics.
The manual for the packages graphics and graphicx gives the basic information about importing graphics in LaTeX. The Strategies for including graphics in LaTeX documents article by Klaus Höppner gives a basic introduction and overview to graphics in TeX.
Hypertext. The source of the url LaTeX package describes how to handle the special needs of typesetting hyperlinks. The hyperref package does its best to convert your LaTeX document into a web-optimized version.
Presentations. The user guide for the Beamer package is extensive and detailed. A more gradual approach is in the Beamer by example article by Andrew Mertz and William Slough.
Editing and running: TeXworks. The TeXShop editing system is Macintosh-specific. The TeXworks editor is cross-platform, and was inspired by TeXShop. It tries to include the most valuable features of other editors while retaining simplicity.
Fonts. The article discusses fonts that are still being developed for use with mathematics. More on the topic, including links to the latest on those fonts, is at the TeX Users Group font page. A comprehensive list of the variety of choices available today is A Survey of Free Math Fonts, by Stephen Hartke.
Obtaining TeX. To get or upgrade your computer's TeX system you can use the distribution appropriate for your platform.
All of these distributions are free software. The TeX Users Group maintains an extensive list of resources, a complete list of distributions, both free and proprietary.
Best practices with publishers. The AMS author FAQ from the American Mathematics Society contains many tips that apply when working with the AMS, and also tips that work in many cases.
TeX is free software, and exists for and because of its users. Please don't hesitate to come forward.
One way is to contribute software or documentation, if you have the skills. For instance, you may notice a job that needs doing when you look for something for your own work and cannot find it;—perhaps you are writing a paper for a conference and have worked out a LaTeX style file. If you do this, please don't stop at putting it on your web page. Take one more small but very important step and contribute it to the community's archive where it will not disappear or move its web location when you get a new provider. Furthermore, TeX Live and MiKTeX use CTAN as the reference source.
Another way that anyone can help is to join a user group. Most groups have a publication and regular meetings. Most also sponsor development of the software, so simply by joining or having your organization join you are making a significant contribution.
Some things were mentioned in the article but not featured. For your convenience, they are listed here.
The article's LaTeX source is here. It is released under the GNU GPL license: you can do with it as you wish under those terms, including the graphic. (The article as printed in Notices uses the Lucida fonts. The source here has been modified to use Computer Modern.)