You want to have the highest bid on some other player before the bar fills all the way up.  After they've launched, it won't help to keep adding to the bounty on someone else.  Pick on one enemy that you want to place the bounty on - you DON'T want to spread it around.  As a side note, the bounty is something that all players will have access to - depending on how much mayhem and destruction any player causes, they'll draw from the bounty.  Because that's the only way pirates get paid, that's all they pay attention to.

If someone stuck a last minute bid on you (a common occurance, so I usually wait to add bounty right up until the last moment)  that you don't have a chance to overcome, start building defenses and ships at the closest planet to the pirates.  In the beginning of the game, the pirates aren't too terrible to face as long as you have a decent defensive fleet and you've built up your planet (speaking of which, purchase at least the first upgrade for planet infrastructure so they have more hitpoints).  Personal preference: if I focus my first strike on the pirate planetary assault ships, I find it becomes much easier to deal with the rest of the vessels.

Also, one thing I've noticed about pirates is they only go after human players if no one places a bounty.  If you are a new player and not placing any bounty, it could be quite frustrating having to deal with a pirate attack every single time.  If the pirates are too much of a problem right now, try turning pirates off when you start a new game. If you don't want to be hassled at all and just want to get a feel for the game, you can even turn off AI players and play solo for a while (though you might want to leave pirates on if you do this...so you have a little combat now and again).

After you get the hang of the game, do what Pignasty said: farm the pirate attacks for capital ship XP (personally, I leave pirates off for this reason...it makes the game way too easy).

Another option is to build up a fleet pretty quickly.

All that's really necessary is a few Repair Bays and a capital ship to ward off pirate attacks. Just keep the cap within range of the repair bays and it'll start accumulating experience.

Pirate Cutthroats and Rogues (their light frigates) will invariably give trade ships priority - they will ignore everything else to chase down these ships. Sometimes these ships are leading the raiding group, so you may see Corsairs (the Pirate HC) following them, although they tend to go after your orbital structures instead, giving tactical structures priority (they tend to attack static defense, repair bays, and sometimes hangars). Reapers (Pirate flak) will generally follow the Corsairs. Pillagers just charge for the planet just like any other siege frigate.