A puzzle I've played that I still remember vividly is the Rubik's Cube. I got hold of one during my second year at university, when a friend left it behind in my room after a study session.
For anyone who hasn't tried one, it's a plastic cube made up of smaller coloured squares, and each of its six faces can be twisted. The idea is to mix it up and then get every face back to a single colour. It sounds simple enough, but my first attempts were hopeless. I'd manage to complete one side, and then the moment I tried to sort out the next one, I'd ruin everything I'd just done.
I found it incredibly difficult at first, mainly because I was relying on luck rather than any kind of method. Eventually I swallowed my pride and looked up a beginner's method online, which breaks the solution down layer by layer using a handful of set moves. Even then, it took me about a week of practice in the evenings before I could solve it without checking the instructions.
As for how I felt, it was a real rollercoaster. There were moments when I genuinely wanted to throw it across the room, but the first time all six sides lined up, I was absolutely thrilled. What I liked most was that it rewarded patience more than cleverness. These days I can solve it in a couple of minutes, which isn't fast by any means, but I still find it oddly relaxing.