When I install a program like GIMP or LibreOffice on Linux I'm never asked about permissions.
These applications are installed in a privileged part of the file system which you and most users don't normally have access to change.
On mainstream distros set up for desktop use the single user initially set up when installing will typically have admin rights. All that they'll normally be asked for is their own login password to install software, and not always that.
Once installed the software will, by default, be setup to be executable by normal users and allow data files to be read. That's all that's needed.
By installing a program on Ubuntu, am I explicitly giving that program full permission to read/write anywhere on my drive and full access to the internet?
Not the program. What happens is that the user account belongs to different groups and the different groups grant access to different resources.
The security model on Linux is that your user account has certain rights, and the groups your account belongs to have specific rights. To right to any part of the file system requires root privileges, which are normally not granted to users. Even when you use sudo you do not get all rights.
Normal user accounts typically have access to the resources they'd be expected to need, like the internet.
Theoretically, could GIMP read or delete any directory on my drive, not requiring a sudo-type password?
Any directory or file you, as user, have rights to access can be accessed by the application you launch as it typically inherits your rights. A normally logged in user will not by default normally have the right to alter most of the critical system files or shared files.
I'm only curious if it's technically possible, not if it's likely or not. Of course, I know it's not likely.
Keep in mind that most users typically install applications from repositories that are well policed and the risk of hostile code is low.
I'd probably suggest some additional reading to get to grips with Linux permissions.
The Android security model is evolving to match the needs of users who not only don't generally understand anything about the underlying security model of the OS but hardly understand anything about computers.
The Linux model (which I frankly like more) is designed to allow users (and administrators in particular) to exercise more control over security on the system (within the limits if their allowed permissions).
I think from a user standpoint it's best described as the difference between full-auto and semi-auto or manual control. Modern consumers want full auto. Linux has semi-auto and manual. Most Linux users never need to know about the security model these days, but the control is there if you need or want it.