Impossible to say unless you tell us what qualities the best weapon should have. Also do you mean beams, turrets, cannons, missiles, etc. or do you mean phaser, disruptor, polaron, tetryon, antiproton, plasma, etc? Phasers seems to be a popular choice since it can disable a particular system while antiproton does more damage when it crits.
There is no such thing as "best weapons", "best ships", "best career", etc. It is all about how you set up your character. Read the descriptions for weapons and pick the one you like. Once you've done that stick to only that type of weapon and increase the skill level of your character for that particular weapon. For example, I use tetryon energy weapons so my character's tetryon weapon damage skill is at the max. You just need to pick one and stick with it, you could always change later on.