It's something I always have never been quite sure about... I usually go with drivers first, updates and then programs, but maybe it doesn't even matter at all...
Updates before drivers, because if something doesn't work it might be an outdated driver that is already installed, and an update will fix it, whereas downloading a driver from the internet might cause a conflict with the currently installed one or with your hardware if you download the wrong driver.
Whenever I install a new operating system my monitor's resolution is all messed up, and by updating the system the driver gets updated and the resolution adjusted. On the other hand I see a lot of questions here from people with fresh installs that download proprietary drivers on Linux as soon as they boot the system and see that something isn't working, like the display's resolution. The proprietary driver just makes things worst, since the free one was just outdated.
As for applications... I'd install them after everything else since it's the least important part in a "getting things working" point of view. You can install them first, but some might not work with an outdated system (unlikely to happen).