OSX El Captian: How can I switch screen without Apple Magic Mouse or Tracking pad

Use whatever shortcut keys you would like to define for moving between spaces, in System Preferences > Keyboard > Shortcuts. Default should be control+← and control+→, but you can change it.

An alternative method is to define a Hot Corner to toggle Mission Control.

Additionally, note that for most applications, you should be able to hold down option (perhaps alt depending on your key mapping), to override the default full screen behaviour.

I made Mac Mouse Fix.

It lets you remap your Side Buttons to switching between full-screen apps among other neat stuff.

It's also 100% free, it's light on system resources, easy-to-use, and fully open source.