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, are compiled to portable Z-Machine byte-code and as a result they run in 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 most recent interpreters will run games written in the most popular languages.
Some of the most popular interpreters are Frotz for Windows, iPhone & iPad, and Zoom 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.
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.