Fighting the Dreadnought
The Dreadnought is fought twice in quick succession, and whatever paradigm you finished the first battle with will be selected for the second fight. The same strategy works both times, with the only difference being that the Dreadnought will have more health, use Wrecking Ball - a very devistating attack - and use fire based melee attacks in the second battle, meaning at least 2 staggers are likely needed before you win.
When fighting the Dreadnought, a good strategy is to have Vanille debuff the Dreadnought, while Sazh buffers the team (use Bravery / Faith, depending on the character) and Lightning goes in with Commando. Once the Team is fully buffed and the enemy fully debuffed, you should switch to RAV,RAV,RAV and keep pressing the attack until the Dreadnought is staggered. Once done, you can either switch to RAV,COM,RAV or stay as RAV,RAV,RAV to defeat the enemy.
Come the second battle, the same strategy will still work, however, be aware that you will almost certainly need to fall back to a role that includes a medic if and when Wrecking Ball is used. Also, the Dreadnought will also be able to include fire in its attacks, so be ready for taking a bit more damage this time around.
Upon Defeat of the Dreadnought, the Crystarium will be expanded to stage three, Sazh and Lightning will be able to use the Commando and Medic roles respectively, and (possibly most useful of all), you'll now be able to upgrade weapons and accessories, making them more powerful and useful than before!