The barrel is always mentioned a million times over in topics like this, but I had read about it on the internet before I had access to Sonic 3, so hooray for me!
Anyway, what really tripped me up as a kid were two specific instances. I had access to Sonic Adventure from age 5, and Sonic 1 from age 7 (thank you SEGA Smash Pack Volume 1!). Big's entire storyline in Adventure, especially
Hot Shelter stopped me dead in my tracks as a kid. I just found it so frustrating and was constantly questioning why I was fishing instead of doing stuff like the other characters were. It was a wonderful day when I found that stupid fish in Hot Shelter. Tracking down the lure upgrades to get some of the extra emblems made me quite angry as well.
I have no shame in admitting that despite being able to play it since 7, I never actually beat Sonic 1 until I was about 12. Marble Zone tripped me up for years. I was awful at side-scrolling games, but moreso with Marble Zone. A lot of you guys had Metropolis Zone as a kid, I had Sonic 1's second stage.
At the time I just couldn't grasp the concept of slowing down and waiting on platforms that often, I suppose, particularly when Green Hill required little of that practice to beat it. Amusingly, something with the sound emulation was a bit odd in Smash Pack if I recall correctly, so the special stage music was quite different from what it actually was. Perhaps it was something with my TV's audio as opposed to the emulation, but I remember playing Sonic 1 on Kega Fusion and going "WHOA! This sounds completely different!".