How do I use boarding parties? I bring up the weapon's officer, I have an enemy ship selected, I tell them to target the enemy's weapons.... and then I wait? Nothing seems to happen even when I'm right on top of the enemy. Is it just a matter of time/chance? The manual references transporting in marines which suggests that I need to bring down their shields, but this is the tutorial mission with my weapons disabled. So how do?
You can only transport marines to a ship that's facing you with a section that has a lowered shield. This means that you must in some way drain a shield on one side (by firing, throwing mines, etc.) or be "lucky" enough to have them transport a mine near you (since transporting lowers the shield in the appropriate direction temporarily). Only then can you transport marines to them. You can give the order beforehand and once all conditions are met, marines will be transported automatically. I think it takes two tries to destroy a weapon on an enemy ship.
EDIT: Missed the bit where you mentioned the tutorial mission. I just gave it a quick run - the trick is in (transporter) mines - a single mine detonation is enough to crush a shield with a single red line. Also, once I tractor beam'd the enemy ship, all of its shields lowered somehow.