I have a Tentacruel with both quick and charge poison moves. As a water type it's weak against grass, but it's poison moves are strong against grass. Does this make it neutral for grass types or does type take precedence?
Tentacruel is poison/water, therefore grass will do regular damage to it because it is super effective against water, and not very effective against poison, and the two modifiers will cancel out. However, when you attack using poison type moves, it will be super effective against something that is just grass, and it will also benefit from STAB (Same Type Attack Bonus), so it will deal additional damage with poison type attacks.
Type bonuses occur from the attacker's move type against the target's types - the target's moves don't matter. To illustrate, using Ember (fire type) against a Bayleef (pure grass type) is super effective, whether or not Bayleef knows any fire, rock, water, or dragon type moves.