Is there a simple way to measure how the upload or download speed of data leaving my computer, like when I'm backing up the computer, or uploading large files through a browser?
If so, how can I do this?
Activity Monitor in the Utilities folder will show network usage, disk usage and many other things. You can sort by process as well.
MenuMeters - sits in your menu bar showing a constant graph and/or numeric values for upload & download on any interface.
You can do it using iStat pro. It shows you how many GB you've downloaded and uploaded since you booted up.
In addition to iStat Pro, Little Snitch will also do this. Little Snitch also provides reverse firewall protection, making sure you know which applications are connecting to the internet.
For El Capitan and after, the wonderful Menu Meters listed above no longer works. However, there's a fork of it available which does support El Capitan and adds some new features.