The type of the Pokemon only matters in terms of defense. The original Pokemon games have STAB (Same type attack bonus) where if a Pokemon used a move that was the same type as they are they would do an additional 50% damage with it. It is unclear whether Pokemon Go has implemented STAB.
So if your Psyduck (water) is hit by a lightning attack it will take double damage.
The attacks of your Psyduck have nothing to do with the type of the Pokemon, so whether they do extra damage or not is based on the type of the attack. So your cross chop for instance will be super effective against normal types.
This type advantage chart for the 6th Generation games shows which types have advantages against other types:
Although note that in Pokemon Go, immunities (0x damage) do not exist and is instead a not very effective (0.5x damage) attack.