First you get ammonia from fish waste, then helpful bacteria in your gravel and filter break that down to nitrite, then another bunch of friendly bacteria break it down to nitrate. Last Monday you had ammonia and nitrites, implying that the first lot had started to multiply. Now you have zero nitrites, so you've got some of the second lot, too, although you've still got some ammonia...
Yes you do need to wait until ammonia and nitrites are both zero before adding more fish. It would be worth getting the nitrate tested as well, though - its presence proves that you've got all your bacteria growing nicely, but you also need to keep an eye on it that it doesn't get TOO high - water changes will keep it down. (Are you adding any water conditioner to the tap water?)
So yes it looks like the tank's nearly gone through its first mini-cycle with the fish you've got. But build up your stock very gradually, as every time you add fish, you'll get another mini-cycle, as the bacteria multiply to cope with the extra waste. So test your ammonia/nitrite whenever you add fish, and don't add any more until they're back to 0 again.
I don't understand what they said about the filter cartridge; that's where most of your bacteria live. For the same reason, never clean it out under the tap - always use old tank water.
It sounds like you've had a rough start but it's now coming together for you
Again, do read all the stuff on this site, there's so much info here! Good luck!