Why R?

Should I use the R statistical software in my introductory statistics course?

I assume you know that statistics is done with software and that learning a reputable software package should be a part of any applied statistics course, so that the question is which software.

Advantages

R is free. In addition to helping your school's budget, this means no whining for money, and immediate installation on any machine under your control It also means students can take it home and install it on their own machines. High school students can take the software to college with them.
R runs on Windows, Mac and Linux/Unix platforms, so it probably runs on the hardware your school has, and the hardware your students have at home.
R is professional quality software used by professional statisticians. This sets it apart from Excel and various education programs.
R includes a vast array of statistical procedures. Few of your students will ever outgrow it no matter how many statistics courses they take. This sets it apart from graphing calculators which are a terminal technology supporting only the first course.
R was written for advanced graphical data analysis and has knock-out graphical output.

Disadvantages

The usual documentation for R assumes you already have a Ph.D. in statistics and extensive programming experience. However, you do not need that to use R, only to read the documentation!-) I needed 11 handouts for my intro. course. You can use them to learn R or for your own students.
R has a command-line interface. There is an optional graphical interface called R Commander. There are two handouts I made for a statistical literacy course to help you get started with R Commander, one for measurement data and one for categorical data. The presentation assumes you read them in that order.
There are not a lot of people using R in introductory courses so R-specific resources may be few.