Lets see...
When leveling, my advice is to fly an escort (if you can get the dual heavy cannons)
Space
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Weapons:-
Generally for energy weapons, the smaller their firing arc, the more powerful they are.
As far as energy weapons go, dual heavy cannons are best with regards to damage. While the tooltip says they do the same DPS as a dual cannon but require more power, dual heavy cannons actually do more damage because they fire less often and as such use less power. Bit confusing I know.
Common odd marked dual heavy cannons I believe are only purchasable on the KDF side. If you want some, just give me a yell and I'll have my KDF captain buy some for you.
For the federation, these are generally escort only weapons. For the Klingons, these can be equipped on all their battlecruisers, escort-type ships and carriers (which is a mistake, given the poor turnrate).
Energy weapon types and their procs:-
Phaser - Small chance to disable a subsystem (like engines, weapons etc).
Disruptor - small chance to debuff damage resistance.
Plasma - chance to cause "damage over time" (DOT)
Tetryon - small chance to damage additional shield health.
Polaron - small chance to decrease targets power.
Antiproton - higher crit damage. Only available late in the game.
Due to attack pattern alpha being able to boost crit chance, antiprotons are generally best for tacts. Also, since you want to biggest bang for a shot taken, dual heavy cannons are best.
My advice is to go for either phaser or disruptor. Antiproton if you a tact captain using dual heavy cannons or torps.
I prefer to use Tetryon, not for the proc (which is pathetic), but because most PvPers seem not to be carrying specific resists for them.
Torpedos and types:-
Photon - Basic torp.
Quantum - longer firing rate, but flies through space faster and deals more damage.
Plasma - pretty much the same as plasma weapons.
Transphasic - deals an extra bit of shield bleedthrough damage. However since it's damage is so poor, it's really not worth it. The special transphasic cluster torps are equally as useless.
Chroniton - Has a 33% chance of slowing the target, which is pretty much the only reason to take them.
Hargh'peng - Possibly the best torp in the game. Isn't affected by torp abilities, but in addition to dealing normal kinetic damage, it'll also inflict a radiation debuff (damage over time). Finally, when the debuff timer finished counting down, there will be a secondary explosion that ignores shields.
Tricobalt - Deals a lot of damage, but has a long cooldown and can be targeted.
Ships:-
When levelling up your character, my advice is to fly an escort (if you can get access to dual heavy cannons). With an escort, the space sections will be over in a flash. Unless your doing PvP or an STF, tanking in this game is easy, so you'll run into little difficulty in what is supposedly the squishiest class in the game.
Unless you're attached to the federation cruiser ships and/or intend to be a healer in PvP, I have a hard time recommending them. The game has them basically designated as healers/tanks. While klingon battle cruisers can deal damage effectively (if using dual heavy cannons), Fed cruisers lack bite in comparison. Don't get me wrong they still can wipe out NPCs easily enough, but given how easy tanking is in PvE (exception being STF's and maybe elite PvE), this makes them a bit worthless in comparison to an escort.
If you're still intent on being a damage dealer while flying a federation cruiser at level cap, here are the cruisers I'd recommend flying:-
Assault Cruiser (Sovereign) - Non c-store
Better turnrate then a star cruiser, has an extra ensign tactical station.
Advanced Heavy Cruiser Retrofit (Excelsior)- C-store
Best turnrate of any tier 5 Fed cruiser. Unlike other Fed and KDF cruisers, it has a Lt Cmdr tactical station (allowing it to use tier 3 tactical bridge officer skills)
Dreadnought (Galaxy-X)- C-store
The only Fed cruiser that can equip dual heavy cannons. However, due to it's poor turnrate (worse then the sovereign and excelsior), it's not recommended to use dual heavy cannons in PvP. It also has a phaser lance attack, which is basically a massive cannon you can fire off every few minutes. One of two fed ships with a cloaking device.
I'd recommend against getting the exploration cruiser refit (Galaxy refit). The saucer separation skill is more or less a gimmick and if you get destroyed the skill is put on cooldown.
As for the KDF, their best cruiser is the Vorcha refit. Same bridge officer stations as a sovereign, lower hull, but has the best turnrate for a cruiser in the game at tier 5.
Ground
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Not much to say here.
Best everyday weapons are generally the pulsewaves(low firing rate, high damage as well as a possible knockback).
I suggest you have at least one science bridge officer skilled as the away team healer.
VS Borg:-
Borg adapt to your attacks after a few shots, so you'll need a remodulator to clear the debuff (which can be replicated by your ships replicator). However, since the normal remodulator takes 4 seconds to, it's best to have a weapon that deals high damage with the least shots (basically the pulsewave for the most part).