What was the 'quantum mechanics' before quantum mechanics? Quantum mechanics is near-universally considered one of the most difficult concepts to grasp, but what were the persistently unintuitive, conceptually challenging fields physicists had to grasp before the emergence of quantum mechanics?
The aim of asking this is, mainly, to gain insight on how one may approach a subject as unintuitive as quantum mechanics.
 A: Influence at a distance. 
This is commonly how gravity in Newtonian gravitation was understood. Newton himself described it as anti-intuitive and could not see how any man trained in natural philosophy (the physics of his day) could accept it. 
In fact, this notion goes much further back. Over two millennia ago, Aristotle described external force as that which can cause a change in the natural motion of another and this by contact. 
(He also had a concept of Internal force. This causes internal growth and change but there is no notion of contact here. This is implicit in Dylan's poem 'The force that drives the flower...' but I very much doubt he was referring to Aristotle's theory explicitly).
Yet Newton's theory had exactly this. It was accepted because of the success of his theory a posteriori in explaining many many things. Its very success hid its conceptual problematics, but there were people who struggled over it looking for a mechanism that would explain gravity in a local manner.  This is at the root of that forgotten concept called the aether and there is, in fact, a long history of such attempts if one cares to look into it.
The solution to this anti-intuitive mechanism was finally found three centuries later. First Faraday discovered the field concept, and this was applied to Electromagnetism by Maxwell. Then it was taken up by Einstein in his theory of gravity. He identified the field of gravity with spacetime itself and this, understood correctly, finally made sense of that stillborn concept, the aether. In fact, where physicists had gone wrong with the aether was to try and conceptualise it mechanically. The field was not mechanical, and much more flexible. It remains a pervasive concept in modern physics and much elaborated upon. 
It's chastening to think a similar time-scale might apply in sorting out the puzzles associated with QM. In which case there are two centuries to go! And one is very much aware of a tangled history of attempts to make conceptual sense of QM. To turn it from an operational theory to an ontic one - we would very much like to know what is there and not merely operate an efficient machine that tells us the answers to our questions.
Another example might be imaginary numbers. This, however, was the discovery of mathematicians rather than physicists.
A: Taking 1900, the date of the formulation of Planck's blackbody formula, as the birth of quantum mechanics
Electromagnetism and statistical mechanics.
These were the topics where the cutting-edge research in physics was done at the time. The most brilliant theoretical physicists of the 19th century (James Clerk Maxwell, Hendrik Lorentz, Ludwig Boltzmann, Josiah Willard Gibbs...) were doing research in one of the two or both.
Both fields require a mastery of non-trivial mathematics: vector calculus and quaternions (1) for electromagnetism, combinatorics and probability theory for statistical mechanics.
Also, both fields contain deeply counterintuitive ideas. 
In electromagnetism: the (absolutely non trivial!) relation between electricity and magnetism, the discovery that light is nothing else than a wave in the electromagnetic field, the fact that this wave was able to propagate through empty space...
In statistical mechanics: the fact that it was possible to obtain physical laws by applying probability theory to large systems, the connection between the time-reversible microscopic dynamics and the irreversibility in the macroscopic world, the idea itself that matter was composed of atoms and molecules, which was not at all widely accepted in the scientific community of the 19th century.
Quantum mechanics and special relativity were born from research in one of these two fields: think about the blackbody problem (statistical mechanics), the research on the "luminiferous ether" that led to the Michelson-Morley experiment, Einstein's famous paper On the Electrodynamics of Moving Bodies...

(1) Quaternions, an extension of the complex number, were much used in electromagnetism at the time.
A: Quantum mechanics certainly deserves its reputation as non-intuitive. It is certainly possible to build intuition in quantum mechanics. Other fields of physics lend themselves to analogues, spatial reasoning, or thought experiments. One of the only ways to build intuition is through the mathematics of quantum mechanics. 
As far as studying the development of other highly non-intuitive sciences (I hope you don't mind ones that post-date rather than pre-date the invention of QM), there is some good reading in the following areas:
Nonlinear systems
Have you read Chaos by James Gleick? He gives a good historical account of how hard it was (and still is) to break down the philosophical barriers to the study of nonlinear systems; specifically  how a simple system can exhibit complex behavior when even small nonlinearities are introduced. For a more mathematical account, Nonlinear Dynamics and Chaos by Steve Strogatz is a classic. 
Games
It's not a field of science, but some researchers (Nature 532, 210–213 (14 April 2016))recently invented a computer game called Quantum Moves you can play to help find solutions to quantum systems. It's the closest interactive visualization to time-dependent quantum systems I have seen.
A: 
The aim of asking this is, mainly, to gain insight on how one may approach a subject as unintuitive as quantum mechanics.

Everything's unintuitive until it isn't. Your profile says you're undergrad math and physics, so you'd probably agree that Newtonian gravitation is pretty reasonably intuitive. But historically, that just ain't so. Europeans, particularly the French, found Newton's gravitational theory entirely unsatisfactory when he first published it. They wanted to know what gravity is, whereas he just described how it behaved. But what is it ? And Newton never answered that to the satisfaction of his critics. And we basically still can't satisfactorily answer that, but we've learned to accept descriptions of Nature's behavior as intuitively satisfying, without needing to reify (make a "thing" out of) the abstract mathematical descriptions.
So now quantum mechanics comes along, begging you to ask, "What is a particle ?" And then providing what seems like a totally unsatisfactory unintuitive semi-answer. At least that's how it seems today. But qm does provide a so-far-unchallenged accurate description of a particle's behavior, whatever the heck it is.  As far as that is part goes, it's a "correlated bundle of observable properties", but not all of them simultaneously measurable (the bundle has a collection of maximal subsets that are simultaneously measurable, which Schwinger called "complete measurements").
Moreover, consider high-energy collider experiments, where if two, say, protons collide with sufficient energy, then any kind of particle whatsoever can emerge from that collision (as long as there's enough energy in the collision to create it). So if that particle were something, so to speak, you wouldn't intuitively expect it to just pop into existence out of nothing (i.e., nothing but enough energy). But that's what happens, whereby "correlated bundle of observable properties" may be a better way to think of it. In any case, over enough historical time, what seems unintuitive today will eventually come to seem natural and satisfactory and intuitive. (Or so I'd guess:)
