To start, lets mention the most general (and perhaps obvious) shortcut keys on the keyboard which will help you increase productivity on a daily basis if you learn to adopt them. Here is the full, official list provided by Apple for keyboard shortcuts and here are the shortcut keys for Windows users.
By law, all operating systems must be fully functional without using a mouse. This is thanks to Universal Access rights for people who are only able to use a keyboard. Knowing this, we can figure out how to enable what is called Mouse Keys on a Mac or a PC. Mouse Keys is a program which allows you to use the numeric keypad to adjust the motion of the cursor as you would a standard mouse.
PC: Navigate to the Accessibility Options in the Control Panel. Once there, select the Mouse tab. There will be an option to select "Use MouseKeys". Once enabled you can also access the MouseKeys settings from the same tab. Microsoft also provides an efficient way to enable MouseKeys if you don't have a mouse in the first place.
Another (more simple) method of using your computer without a mouse is to transform your smartphone into a wireless mouse! There are actually numerous applications which allow you to do this.
iPhone: We like the iMouse app (despite its terribly unoriginal name). The interface is simple and it does exactly what you would expect it to do to. In order to make this work, however, you need to download the software from their website first. Once the software is installed you must enter your IP address in order for your iphone and your computer to communicate. Once setup you now have the ability to click, scroll, and access a full keyboard straight from your mouse.
Android: For the Android user we recommend using the RemoteDroid application. It again has a nice interface and has a straight-forward install along with the same functionality as the iMouse.