The Silver Age Superman was dramatically different from the Golden Age Superman. Aside from the very early stories, the Golden Age Superman tales tended to be more of a whimsical series. Perhaps sensing that Superman was too powerful to face ordinary crooks, the Golden Age Superman generally found himself up against conmen like J. Wilber Wolfingham, or other enemies who used guile and cunning like the Prankster or Luthor.
The other option was to weaken Superman, and this DC, especially under Weissinger's editorship, began pursuing with a vengeance. Although Green Kryptonite had been introduced in the comics as early as 1949, and in the Superman radio series years earlier than that, it had only featured in occasional stories prior to the Silver Age.
In Adventure Comics #255, Red Kryptonite made its first appearance in a Superboy story. Red K had an unpredictable effect on Kryptonians, and what's more, each piece affected them differently, conveniently for plot purposes a period of exactly 24 hours. It would be quite a task to list all the changes that Red K worked on Superman over the years, but a short listing includes making him only able to speak and write Kryptonese, turning him into a dragon, causes him to lose his powers temporarily, and even split him into two people--one Superboy, one Clark Kent.
DC expanded the Kryptonite line to three varieties in Superman #157. Quex-Ul, a Phantom Zone Prisoner, is released and vows to get revenge on Superman for his imprisonment. He has observed Gold K taking away the powers of a Krytonian beast permanently, and knows it will do the same to Superman.
Of course, the problem with Gold K is that being permanent, DC could never use it on Superman, only threaten its use.
In addition to Kryptonite, we learned that Superman had another vulnerability: Magic. This required the return of a Golden Age villain, Mr Mxyzptlk (although in the GA he was known as Mxyztplk), who proceeded to appear with alarming regularity. Several other characters used magic to cause trouble for Superman as well:
We also learned that Superman lost his powers under the influence of a sun that was not yellow, like Earth's:
The funny thing is that whenever the plot called for Superman to pick up the nearest mountain, they'd have him do it without blinking an eye, which just kept up the pressure to weaken him.