I just download the 2.77 tarball for Linux but I don't see a way to install the program. I can navigate to the extracted directory and run Blender via ./blender or ./blender-softwaregl but I don't see an install script. Am I missing something?
No, you don't miss anything. The tarball for Linux as well as the zipped version for Windows has no installation routine whatsoever. Extract the archive to whatever directory you like, create a link and you're set. If it runs fine, there are no drawbacks besides of manual updates.
I started using the tarball version years ago after running into downgrades during system updates quite frequently. I always extract it into /home/username/bin/ and also create my link in /home/username/bin/
Alternatively you can use /usr/local/bin/ which would be the correct directory for packages that aren't managed by the system's package manager.
If you want to use an installed version, search the repo of your Linux Distribution.