There are seven basic stations on a ship: sailing, carpentry, bilging, gunnery, navigation, radar and the crow's nest. The first three are often referred to as the "basic duty puzzles". Sailing propels the ship. Carpentry prevents and repairs damage to the ship. When the ship is damaged, water accumulates in the bilge and slows the ship down; bilging is used to clear out this water.
The right-hand side of the screen shows three status indicators for the ship. The first, yellow one, shows how fast the ship is going; the more yellow in the gauge, the higher the speed. The second, red one, shows how damaged the ship is; preferably, this gauge should be empty, indicating no damage. The third, blue one, shows how much water there is in the bilge; preferably, this gauge should be empty or with just a sliver of blue.
When you first come aboard a ship, you should say, "Ahoy", and ask the commanding officer what station he'd like you to take. Most commanders will give orders in chat. Sometimes they'll be specific, telling certain pirates to do certain things. Many are often pretty vague, though, saying things like, "One carpenter, rest on sails," which leaves it to individual pirates to figure out where they want to go. Some commanders will use the order interface, which sends you a whistle and a note in red telling you what you should be doing.
All ships need at least one carpenter; the bigger the ship, the more carpenters are required. The next priority is sailors, for speed. Thus, if the commander doesn't give you a specific order, you should take carpentry if there isn't one carpenter already, and a sail otherwise.
If the ship is damaged and the bilge is high, the commander is likely to prioritize fixing the damage over everything else, so if there's a lot of red in the gauge, unless you're ordered otherwise, you're likely to be wanted on carpentry. Afterwards, the commander is likely to order some people to bilge until the bilge is clear.
In general, you should never bilge unless given permission to do so, and as soon as the bilge is clear (the blue status gauge is empty), you should ask where you're wanted next.
The remainder of the duty stations are specialized and optional. Gunnery loads the ship's cannons. Navigation acts as an accelerator to sailing. Radar allows a long-range view of ships in the vicinity. The crow's nest allows a better view of the ships that are very near, thus making it easier to intercept or run away from them.
If you're a respected pirate in your crew, you can ask if you can load cannons or Navigate; the commander may grant your request. Otherwise, you're likely to only perform these other duties when the commander specifically orders you to do so.
The duty puzzles are scored on efficiency, not speed. However, your overall effectiveness rating is based on how much you've accomplished within the given timeframe (over the league, or the like). So you can't be too slow. Score sampling is done in ten-second intervals, and your current performance is based on the average of three samples.
Thus, for instance, if you make no score at all for thirty seconds, and then you pull off a gigantic high-scoring combo move, your score for that thirty-second period is that combo move, divided over the thirty seconds. This may turn out to be a lower performance rating that if you had made a number of moderately-scoring moves during that thirty-second period.
Because of this time factor, you cannot sit forever and think about your next move. Nor can you spend a lot of time chatting. Though you can pause a duty puzzle by using the Escape key, and type into your chat box then, the time you're spending talking is time that you're not making any score. Thus, pausing the puzzle will result in a lower effectiveness rating.
Sailing is the most fundamental of the duty puzzles. The majority of pirates aboard a ship will be on sailing duty, so this is a puzzle to practice and get reasonably decent at, if not extremely good at.
Outside of battle, the performance of sailors contributes to the ship's speed, as represented by the yellow half-circle dial on the right-hand side of your screen. The dial starts out black, indicating that the ship is stopped, and fills with yellow as the ship builds up speed; when the dial is completely yellow, the ship is at full speed. Even when the ship is at full speed, sailors need to continue delivering good performance at sailing in order to maintain that speed.
In battle, the performance of sailors generates tokens for the navigator to use in the battle navigation puzzle. The navigator needs tokens in order to move the ship, turn the ship, and fire the ship's cannons. The more tokens he's got available, the greater the flexibility he has in determining what moves to make. If he's short on tokens, he might not even be able to make the moves he wants to make.
You score when you clear a target. Bigger targets score more than smaller ones. Clearing five in a row or more also generates a good score, but not as much as clearing a target. Four-in-a-rows score minimally, unless part of a combo chain. The fixed, square blocks don't score any more than normal round pieces. Clearing things as part of a combo multiplies all scores that were part of the chain by the size of the combo (double, triple, and so forth). There's a difference between a combo, which is a chain of effects, and simultaneous clearing. Simultaneous clearing scores more than if you'd just cleared one thing, but much less than a double. For instance, clearing two columns simultaneously is marginally better than clearing one column alone, but not as good as a double of colmns, or a single target clear.
Sailing performance is based upon your score over time, and the number of pieces that you've gone through in that time (fewer pieces is better than more pieces, so use your pieces efficiently). You should watch both the puzzle performance indicator (the one that turns gold when you're doing well), and the performance ratings given at the league points, to determine how well you're doing.
Outside of sea battle, your combination-setup vs. performance rating is largely your own problem. During a sea battle, however, you have a responsibility to generate a steady stream of tokens for the navigator. Doing things that would normally generate a high score will generate tokens; in general, you want to perform combos, clear targets, and clear five-in-a-row or more. There is no one-to-one ratio between your score and token generation; there is a slight delay before tokens are generated, as well. You can both obtain a good sailing rating for yourself, and give the navigator tokens, by performing a steady stream of doubles while clearing targets quickly.