I can easily adjust the level of brightness with the function keys while I'm actively working in the OS, but it resets itself back to full brightness after every reboot which, of course, is something I don't want. I would like to be able to have my latest adjustments automatically saved for me so that I don't have to tinker with them every time I boot up my PC.
Version of Ubuntu installed is 14.04.3 LTS
You can achieve that using xbacklight
Then you can check current brightnes by
xbacklight -get
and set it byxbacklight -set xx
where xx is percent... To make it permanent you can make a startup script. here is a great guide how to do so: How to run scripts on start up?