Any guns other than Lasers or Mass Drivers will do for beating the game. Meson or Tachyon guns fire fast, medium-power shots with quick refire and low power drain (Tachyons are moderately better but Meson will do fine if you are short on cash). Particle or Ion guns fire more powerful bursts with lower refire rate and higher power drain. Plasma or Neutron guns fire slow-moving, powerful shots that drain a lot of power. My personal reccommendation is to go with Mesons or Tachyons for the turrets in the Galaxy and a pair of heavier guns for the forward set.
Also, you should have at least Level 2 shields (with the accompanying Level 2 engine upgrade--you need to have the engine upgrade to power the shields and still have enough power left for weapons). Upgrade these to Level 3 once you can afford to.
An important, cheap investment is to invest in Tungsten armor. For a mere 2,000 credits, you get twice the boost to your armor strength that the Plastisteel armor gave you. This can save your bacon in a pinch.
As Ridgerunner suggested, buy an Afterburner. On the Galaxy it boosts your speed from 300kps to 700kps (alot slower than Confed military ships but still valuable in a fight). The best thing about afterburners in Privateer is that they are powered from your main battery, which means that you'll never run out of burner fuel! Of course, if your shields are running at full power as well, you will drain your battery after a short while, but it will recharge as soon as you drop back to normal speed.
Now as for missiles, you want to put launchers in both of the forward slots. Put the tractor beam in one of the turrets--they can't hold launchers anyway. Depending on your personal preference, you can use either Missile launchers or Torpedo launchers. Each has its own benefits and drawbacks.
Missile launchers can carry Dumbfire, Heatseeker, ImRec, or Friend/Foe missiles. Skip the Dumbfires--they have a lower explosive yield than the others. With two launchers you can fire off a salvo of two missiles at once, enough to destroy a Talon in one go.
Torpedos are essentially like dumbfires, but much cheaper, and they have about TWICE the explosive yield of dumbfires (which is to say, about 1/3 more than IR/FF missiles). Since they do not lock onto targets, you must use them like dumbfires, but a salvo of two of them is enough to completely destroy most lighter fighters, and can even kill heavy fighters if you use your guns to wear down their shields first.
In summary, you want to get the strongest loadout you can afford--heavy shields, armor, and weapons. DO NOT try to go out with Level 1 shields and laser cannons--you will be dead meat if you do. Instead, make sure you have enough cash for some decent upgrades before trading in your Tarsus.