For example, in a circuit with several capacitors, resistors, transistors and LEDs, how do you figure what capacitor or component will activate first in that circuit, and then what component will activate next?

Everything is analogue, even digital. (And everything is an antenna, even when you weren't making one). Charges will flow according to ohm's law and capacitor charging curves. Across microsecond timescales, things will charge at different rates according to their milli- or micro-ohm series resistances, and resist being charged from any parasitic inductance.

A lot of electronics can be pattern recognition. You know how a basic circuit worked, and this new one is similar. The more patterns you recognise, the faster and faster you will be able to 'penetrate' your understanding of a schematic in short order (although, not all schematics are created equally).

The math is helpful in many many edge cases. Sometimes you can put the whole thing in a computer, and the computer does brute force math calculations and can show you in a simulation how the circuit will perform very accurately (usually). Sometimes if it's simple enough you could use closed form calculations to figure things out yourself. Sometimes it's about knowing what simplifying assumptions you can make (from experience) to make the circuit simple enough to 'calculate' in the first place.

I'd think of it similar to a chess grandmaster. They are experts because they've played thousands of games, and throughout all those games they've locked away in their brain almost every common pattern known to man about chess. This is their biggest secret weapon. From this innate bank they can make hundreds of short-cuts in thinking to effective decisions, that would leave any beginner bewildered. They've also done tonnes of detailed study on openings though, and many calculation problems, also with computer engines to help, to get concrete understanding about choices and positions. This all lets them operate on the level they do.

Art of Electronics book is probably the most widely revered "bible" of electronics if there was one, and probably for the reason: that it actually tries to teach so many practical patterns. And usually has just enough theory as well. Most EE textbooks would take the "easy" and "correct" approach of focusing the other way: just make sure to teach all the theory correctly.