I would imagine that TARDIS use is not an automatic privilege of Time Lords, or else the Doctor wouldn't have needed to steal one. He's definitely a graduate of the Academy, and all Time Lords are nobles by definition (though Danny Pink's the only one who ever overtly pointed it out, as far as I recall). But he clearly didn't get his own TARDIS.
Given that Time Lords had a non-intervention policy, and that the Doctor fled Gallifrey specifically because he felt they were too insular and uninterested in the outside universe, I'd imagine that TARDIS use is anything but routine. There's probably a finite number of them used only by those who have reason to use them, like government officials, the military, the intelligence agency, historians or scientists, etc. And I'd imagine a lot of those groups don't fly the capsules themselves but hire trained pilots to take them where and when they need to go, like chartering a jet or a helicopter.