Ah, so you are doing a fish-in cycle. With this type of cycle, you have to test every day and do a water change whenever ammonia or nitrite are above zero. The tank will be cycled for the current fish when both ammonia and nitrite stay at zero for a week.
After that you can add more fish. Since you have 3 neons now, you need to add more, though having been through a fishless cycle the current neons won't be in the best of health as they are not hardy fish nowadays. It is safe to add more fish equivalent to a third of the fish in the tank now. Strictly speaking this means just 1 neon, but adding more one at a time is not good practice for shoaling fish so I would get another 3 and be prepared for a spike in ammonia and nitrite. Test every day and again do water changes every time either reads above zero.
You don't say how big the tank is, or what fish you want to get, but with a fish-in cycle you need to take things slowly. Adding too many fish at once will cause ammonia to increase a lot and it takes a while for the bacteria to catch up.
Re the algae, if you get live plants this will help combat algae as well as making the fish feel safer and taking up a lot of ammonia. Plants don't turn ammonia into nitrite either. Floating plants are best for taking up ammonia and they make the fish feel safer. They don't know there isn't a bird lurking just out of sight waiting to pounce and floating plants give the fish something to hide under.
