In all honesty, for me, I didn't move on to anything.
The N64 was the first console I had ever been able to buy myself, in late 1998, before OoT came out. I had had a 2600 in my very early childhood in the 80s, and later "grew up" with my NES, and got hand-me-down SNES by Xmas 1995. But the N64 was the first system I bought with my own money, saved up from a summer job.
I had given some thought to getting a Dreamcast in 2000 or so, because of games like Ecco, and MvC2, and Soul Calibur, and the sadly cancelled Castlevania game that I was interested in. But I never did get one. I eventually go a PS2 around 2004 or so, used for cheap from someone I knew, and gradually started getting a few games for it here or there. I never did actually buy a Gamecube for myself at all, as I had intended to when the GC version of Twilight Princess released, but then I figured "You know what, the Wii plays GC games anyway, so fuck it", and I waited to get a Wii (which I had to wait until Feb 2007 for one to even be available, because they were that in demand in my area).
All in all, that PS2 generation, I kind of "fell out" of gaming a bit. Which isn't to say I didn't still like games, or play them at all, I did. I just didn't really do much in the way of buying new ones for myself. I got Warcraft III and a couple other PC games in the early 2000s, but outside of that, I just wasn't as into gaming that gen, for various reasons. And any games that I was interested in, for PS2 or GC, I usually just checked out at one of my friend's houses. It wasn't until the Wii came out, honestly, that I really started buying games regularly again, though I had bought a DS Lite and a few games for that prior, earlier in 2006.
Overall, as of right now, I do own a pretty decent collection, with an NES (with the most games for that, near 100), a SNES, an N64, a Sega Genesis, a TG16, a PS2, a Wii, a PS3, my DS Lite, and the Wii U. The Wii U being the newest console I own, and only eighth gen one. I may eventually get a PS4 when it goes down in price considerably, and if there are several games for it I actually want, such as, say, if Elder Scrolls VI comes out and is actually good. But I don't know for certain that I'll get one, honestly, as I really don't like the mandatory "have to pay for PS Plus to play online" bullshit. That, alone, and the lack of backwards compatibility, really kind of turned me off to the PS4.