Phew, not a simple question! As an amusing hobby that has gradually grown to encompass more territory, I've increasingly had to at least find storage solutions. The above gives an impression but is not at all comprehensive.
The short version is "just out of sight, jumbled by type" The pinboard above is just one of four, all full, that I have affixed to the inside of some cabinets. I need only pop the doors open to scope things out. The magnetic note board is free-floating and normally resides in a drawer on top of one of my disorganized boxes of postcards (also pictured). Inasmuch as anything is "organized," it's really the decks of cards that I've had to keep careful track of, as their original packaging proved too bulky and didn't provide sufficiently easy access. Those now reside in a carrying case, with decks organized by year to facilitate giving proper credit.
As much as possible, I try to keep things browsable so I can scan what I have and let inspiration strike. Sometimes, it's unguided browsing; other times, I'll have a specific thing in hand already and am thinking, "what would go with this?"
I should emphasize that this reflects a gradual accumulation over time, and any organization that has emerged was simply to save time down the line. There's no grand plan to how things are assembled - I just try to come up with novel combinations ad hoc. Often, when I'm digging around for one thing that strikes me as funny, I rediscover some other thing that reminds of yet another thing, so putting one bunch of items together is usually enough for three or four other bunches to coalesce.