I didn't learn to appreciate Noether's theorem until my second semester in graduate school. I learned the math in undergrad and knew if you did fancy derivatives, you could obtain a conserved current. I thought it was just a way to get the conservation laws for cheap (kind of like how you got the equations of motion from Lagrangians for cheap, as was my understanding at that point).
Then, I learned about trying local U(1) symmetry and trying to make the Langrangian invariant under that and wham, you get EM for free, without even postulating it a priori. I was amazed--all of a sudden, I realized the reason why all these particle physicists were so bent on postulating this or that symmetry, hitherto not seen, might exist in nature; I realized how fundamental symmetries were and how deeply intertwined with conservation laws they were.
Then, I learned about trying local U(1) symmetry and trying to make the Langrangian invariant under that and wham, you get EM for free, without even postulating it a priori. I was amazed--all of a sudden, I realized the reason why all these particle physicists were so bent on postulating this or that symmetry, hitherto not seen, might exist in nature; I realized how fundamental symmetries were and how deeply intertwined with conservation laws they were.