You're entering into a whole new world lol. The quest for the ultimate fuzz/overdriven sound is like the search for the holy grail. There's also a fairly big difference between fuzz and distortion, although some pedals do blur the line.
Fuzz wise I've been having great results with m CoPilot FX Orbit, but thats quite specialist, I've got a Fuzzrocious Grey Stache on the way, but again, quite full-on.
Overdrive/distortion is actually harder I think, the only pedal I've genuinely felt is perfect is my Sonic Kitchen BassBoost Preamp (made by Al Heeley on here), it's a copy/clone of the Xotic BB Preamp....so that might be a good place to start.
It really is about trying as many as you can and deciding from there, it's all so dependent on your bass, amp, cabs, ear etc.
Good luck
Si
p.s.
I have a Boss Bass Overdrive for sale in the FS forum, but that's more of a blendable distortion really. Again, a good starting point perhaps.