It is very important, in any field, to get to the core principles and leave the esoteric stuff for formal courses or for specific applications that may crop up when you need to delve in a bit deeper. That way you will not be drowned in a sea of detail, which 'experts' can try to impress you with. Some of the older books on electronics were particularly bad at this and the smothering effects of verbosity. Once in a while, I have a look at the textbooks that I used in the early days, and they make me cringe.
The other thing is that the field of electronics is so vast that you cannot possibly know it all, but it is important to learn how to learn. That way, when an area crops up that you know nothing about, you have the technique to acquire the necessary knowledge- the internet itself, and the publications on the internet are an absolute godsend in these situations (but you need to able to filter nonsense too). In addition to the internet there are plenty of true experts around who are normally only too pleased to help out (there are some on EEV

). This also applies to math at all levels.
If you work in electronics, very often you will not have any idea what a new project is all about, so you have to acquire the knowledge. One experience I had was a new job, Helicopter Radiometer. I knew what a helicopter was but didn't have a clue what a radiometer was.
To illustrate how not to teach electronics, my introduction to transistors was a drawing of a grounded base configuration, complete with junctions, re, rb, rc, cb, cbe, etc, etc. This was surrounded by a blackboard full of h parameters. And, to cap it all, the instructors kicked off with a long dissertation on leakage currents and the fatal consequences of thermal run-away. Non of this made any sense to the class and was as turgid as ditch water.
But what we should have been taught, initially, is that an NPN transistor has three leads: collector base and emitter and shown a grounded emitter configuration, which is the most common and easy to understand. The initial model then would be that you shove some current into the base/emitter circuit and HFE (current gain) times that current flows from the collector to the emitter. It just so happens that under these condition the emitter base voltage is 600mV. And that is it. Later we found out that the lecturer was actually a mathematician who had recently been diverted to introducing transistors. But don't miss read this- most of our lecturers were fist rate, both at teaching and in their electronics knowledge.
If you get these fundamental models of electronic components fixed in your mind you can go a very long long way in electronics and the whole thing becomes much more interesting and productive. As rstofer says in reply #7, all you need then is Ohms law (simply simple), Kirchhoff's two laws (dead simple), basic arithmetic (simple) and elementary algebra (not terribly difficult with practice) and you have a firm foundation for a life in electronics, if you want to be, a Design Engineer, Development Engineer, Microwave Engineer, EMC Engineer, Lab Technician, Test Engineer, Field Service Engineer, Hobbyist ... Naturally, you would need formal qualification as well for most of these positions.
In parallel with the fundamental learning phase, it is a good idea to specialize in one area that interests you. For many it is power supplies, audio power amplifiers, ham radio, microprocessors, you name it. You would be amazed how much general and detailed electronics knowledge you pick up by this- just talk to a radio ham, about matching, standing waves, Q etc.
The big danger with all this is boredom, but nobody says you have to study all the time. You can still experiment, blow things up and do what you want, but in the background follow a structured learning progression. With this approach I would say that a person with a good logical brain could have a working understanding of the foundations of practical electronics within a year without too much effort, six months if you are reasonably dedicated.
I have said this many times before: I wish to hell that the internet, all the fantastic cheap electronic components, excellent books and videos, were around in my initial years. The other revolution is the low cost, but excellent, test equipment available to you. You can actually get a half-decent digital multimeter in the UK for around £12UK. A DMM wasn't even invented in my day, and a decent analog multimeter cost the earth. And as for an oscilloscope- forget it.

In addition to all that there are computers and applications: ECAD, Simulators, Illustrators, Word Processors, Spread Sheets, Databases, the list goes on and on.
So here endeth the sermon, but just remember that the one most important and overriding thing that is required is you- you must have an ice-cold logical mind, be accurate, terse, consistent, inquisitive and, above all, make mistakes and learn from them.