Here's a link to some serious testing although there's probably been some balancing since it was published is still relevant.
https://forums.frontier.co.uk/showthread.php?t=140240&highlight=weapon+shield+dpe
Essentially energy weapons do the most shield damage but are effective against hull just not as much as kinetics, as I just play PVE I just fit pulse lasers to my ships and put up with the longer time for hull kills.
I've tried a fairly wide range of weapons in Beta testing
Railguns are hugely destructive but with 31 rounds they have to be used sparingly!, they're power hungry and they generate huge quantities of heat.
Cannons are effective but the slow projectile speed makes it very difficult get hits on anything but large slow targets, limited ammo 105 rounds.
PA's do large amounts of shield and hull damage but cost, power, limited (105) and expensive ammo, power and heat make them suitable for big powerful ships only, easier to use than cannons.
M-cannons very effective on hull but need to be used with lasers for shields or you'll run out of ammo fairly quick, low power requirements so useful on ships with small PD's (Cobra 3 and 4?)
Frag cannon essentially a shotgun and really only effective under 500m but can do serious damage to an unshielded hull, limited (93) ammo and medium power requirements.
Lasers come in three types pulse, burst and beam increasing in damage and power drain, pulse is the most energy efficient allowing you to run a smaller power plant and won't drain the weapon cap as fast, again heat needs to be considered as large beams have a much higher thermal load than pulse so avoid on ships that run hot.
Fixed, gimballed and turreted weapons have different damage per second levels pulse is 100%, 82% and 32% so don't use too many turrets although they are very useful on large less agile ships.
Missiles currently have fairly low damage making them unpopular but that could change, they come in guided and unguided (dumbfire

) types and can useful against ground targets.
Mines and torpedos I haven't tried but the single round for the torpedo would only be useful for large high value targets, mines I reckon like missiles haven't been completely developed yet.
Hope this helps!