The first thing a new player needs to work on is the sixth group. The first two are your best active defenses against incoming projectiles; launching them will deflect enemy weapons. The last one is a passive defense that will sit there and repel anything incoming; it can also be deployed actively, but it is less reliable in that capacity. Use it when you're changing positions or if you want to fortify a location.
You can never have too many of those. At least one gravity module is a must for almost any ship build. A not-often-mentioned aspect of repulsors is that scramblers turn them into attractors, which don't have much defensive use but become far easier to destroy.
The best weapons are in the Hyperbol module; the huge, slow bomb and the AOE plasma field thing. To get rid of plasma, you have to launch the black hole thing. I forget the name, but it's in the group to the right.
A cloaked ship with several hyperbols is basically a TF2 spy, able to sneak in and unload at close range for about as close to instant death as you can get in the game. At medium range, HBs can be made exponentially more useful if you detonate a Shockwave right behind them; this gives them a considerable speed boost.
Plasma can also be dissipated by launching repair drones into the cloud. Drones are usually more plentiful than singularities (the black hole things), but they don't work as fast.
You're mistaken about "the group to the right", though. The singularities are in the Impulse module, not Gravity. You're probably talking about the disruptors' ability to collapse singularities. I used to get mixed up all the time about what cancels what, but experience fixed that.
The first group, with the plasma bolt in it, isn't very good. I don't like the second group, either, but I know that it has some of the best damage potential, and the lasers from turrets can't be deflected like other projectiles. Focus on the last four.
The first two groups (ballistics and tactical) are only truly effective if you have a good shot power stat, or your opponent has low shields and armor stats. The Tactical module can be brutal in the right hands, and the upgraded Ballistic weapons are nothing to sneeze at, especially if you have enough capacity to stockpile them. If you're out of disruptors, a stream of red plasma bolts is a scary thing to look at, and the upgraded Cluster Bomb does considerable amounts of damage.
The turrets are at their best in objective mode, where you can plant a bunch somewhere and combine their fire.
I tricked out my ship after logging out today; I sacrificed some deployable and shield strength in favor of maxing out projectile build time and projectile speed. Hopefully, my bombs will be harder to counter.
Sounds like the beginning of an offensive setup; deployable is a common (and perfectly reasonable) dump stat for a direct-fire ship, especially since it seems you want to specialize in hyperbols. You'll need to increase power and/or capacity, however, to make full use of that strategy. Depending on what else you use, area of effect may also be something you can do away with. HBs have an area of effect but it never gets very big.
An unsung aspect of hyperbols is that defensive players hate the upgraded version with a passion. In objective maps, they're basically siege engines; the energy tendrils wreak havoc on wall formations. If the defenders expect upgraded fighters instead, they're likely to have upgraded EMPs available to swat them off, and that's not much use against HBs.
EDIT: Oh, one last note. Try not to run out of Move if you're near someone. Also, EMP shuts down deployables well, but it also stops you from moving; try not to get caught in someone's plasma field.
EMPs also drain your shields over time. Being hit by an EMP sucks hard, especially if it's the first projectile in a stream of regular ordnance.
