First of all, your ship should already be set up for you. Make sure that all your sailors are on the ship. Now, there should be a yellow arrow in the corner of your shipyard (I'm assuming, of course, that you are able to log on). Click it, and if it denies you for any reason then check over your ship to be sure it is entirely assembled (should have guns, and engine, and FCS, and of course a hull, as well as sailors in the sailor slots).

Now, after you've successfully left port, you'll be on a giant grid. Click one of the yellow signs right around the border. You should end up in a different sort of screen, with your ship on a little circle. All you have to do now is simply tab, and you should begin. Controls for the ship, as well as other useful advice, can be found in the manual on this site.

To play things other than the beginner mission, you have to click on a different square (usually they will have a dark-colored sign on them, with a gamename and then a number such as 1/12 below it). Once you enter those, just tab and wait patiently until enough other people join for the game to start.