The first thing you need to do if you want to play some Interactive Fiction (IF) is find yourself an interpreter which will run the games. Many modern IF games, and all of Infocom's works and games written in the Inform language are compiled to portable Z-Machine byte-code. As a result they run in just about any Z-Machine interpreter on any platform. Some games are written in TADS or other languages which have their own interpreters. Luckily, many of the interpreters will run games written in the all of the major languages.
Some of the most popular interpreters are Frotz for Windows, iPhone & iPad, and Spatterlight for Mac & Linux. Zoom, pictured above on the Mac, is particularly nice as it provides an "iTunes-style" interface for browsing games and includes quite an advanced level of customization of typography. And there's also the more recent Lectrote on several platforms.
If you're not all that keen on downloading and installing an interpreter, you can also play many Interactive Fiction titles online on the web. The popular iFiction site has a few Infocom and Scott Adams adventure games that run in your browser. The IFDB Interactive-Fiction Database also has many titles that you can either download or run in the browser.
Here are some links to help you get started.