I've gone ahead and installed Localization Update, Localization Client, Language Switcher Dropdown, and the Language Icons modules. I've also enabled the Locale and Content Translation modules.
I've added two languages to my site (Finnish and French). According to the Translate Interface (Configuration --> Translate Interface), all of my modules are up to date for translations. However, the Drupal Core (7.24 it says, even though I haven't updated to 7.24 yet, still on 7.23), says "No available translations found" for both Finnish and French.
Just wondering if anyone has any idea on how to fix this? I haven't tried manually downloading and installing the .po files yet, as that's not really a solution for me, as I will be unable to do this manually for all of our clients.
Just wanted to add that I've tried this now on a few different setups and I still get the same thing happening on each one. I also tried one machine from a fresh install using 7.24, and still get the problem.