From a classical standpoint, it seems pretty clear that information can be easily lost.
Nope, classical mechanics is reversible. It's so reversible that you can't even squeeze the state space: states that start out "very different" stay "very different". Those vague terms are formalized by Liouville's theorem.
Because classical mechanics is reversible, you can rederive a past state from a future state (and vice versa). Therefore any question you could answer with the past state could also be answered with the future state (just run time backwards for a bit to get the past state, then use it to answer the question). Therefore no information is lost.
In principle, anyways. In practice we don't have access to the whole state and the future state gets so mixed up by the environment that it's utterly intractable to rederive the exact past state from it. Thus thermodynamics.
What set of fundamental [quantum] processes creates or destroys information, or is it somehow conserved?
Quantum mechanics also follows Liouville's theorem. Schrödinger's equation conserves information.
The important exception to this, depending on your favorite interpretation, is measurement.
In Copenhagen, measurement causes collapse and introduces new noise into the system. Collapse prevents you from figuring out the exact future state from a past state, or the exact past state from a future state. The questions you can answer (in principle, with magical access to the whole state and unlimited compute power) are changing over time, so information is no longer conserved.
In Many-Worlds, the "new information" is made redundant by knowing the coefficients of the linear combination of the various decohered outcomes. That allows the past state to be reconstructed from the future state (the whole state, including all the other "worlds").
Another caveat worth mentioning is black holes. People are still trying to figure out how information that falls into a black hole could possibly be conserved, or if it's gone even in principle, or what. See: black hole information paradox.