>To ask a more generic question, where can I go for generalized
>theorycrafting? I find myself with a tac officer flying totally
>unspecced a Heavy Escort Carrier, a sci officer with the Tholian
>Carrier that I carefully specced up 18 months ago and have entirely
>forgotten the logic, and new eng officers in both the RRW and KDF.

Or you can ask here!

I'm actually considering the HEC for the Tac captain I'm levelling at the moment - it's a toss-up between that and the Avenger. Or maybe both!

Here is a great initial, I-just-hit-50-and-can't-afford-anything-serious setup.

Fore energy weapons: 3x Duel Heavy Cannons. I'd recommend the special Tetryon variant that's coming from the Hirogen lockboxes at the moment - their shield drain effect is apparently pretty cool and because the market has flooded, they'll be cheap.
Fore torpedo: 1x Breen Rapid Fire Transphasic (from one of the Breen missions, natch). Good, quick reload, decent punch against shielded targets. Once you rep up with Dyson, consider the grav torpedo - it's the best in the game.
Aft weapons: Tetryon turrets. Hirogen if they're available, regular if not.

Tactical consoles: 4x Tetryon-boosting consoles, as high quality as you can afford.
Engineering consoles: RCS accelerators are always a good fit here - as an escor, your turn rate is very important. Also consider universal consoles such as Phasmotic Leech (expensive as fuck, but best console in the game), Assimilated Console and the Dyson Console. If you get the Assimilated Console, replace one of your rear turrets with the Borg cutting beam and maybe lose your torpedo. Or don't, and run the 2-piece Borg AND Dyson sets.
Science consoles: Usually where I whack universal consoles, or shield consoles if you can afford them. Have you had a look at some of the cross-faction reward pack consoles? There's a lot of fun stuff there for very few ECs.

Shield, deflector, engines: Go with the set rewarded out by A Step Between Stars. Yes, you'll have to do it 3 times. It's also the best free set in the game, AND it's MK XII.
Warp core: The reward core from the Dyson introduction episode (with Worf!) is good enough to be getting on with. Romulans are screwed with this. Poor old Romulans.
Devices: You get a free device from 'Skirmish' (first Devidian episode). It's a 3-minute cool down panic button that boosts your defence. Other than that, batteries or just don't bother.
Hanger bay: A tricky one. First, know that the basic (blue) hanger ships are kind of useless. They start getting good with the Advanced variants from the dil store, and amazing with the Elite versions bought from Fleet holdings. If you can afford Elite stuff (usually costs Fleet Marks and dil) grab Elite Yellowstones. Failing that, Advanced Peregrines are OK, Advanced Danubes are OK as well. There are also Scorpion Fighters available from tiers 3 and 5 of Romulan reputation, but that's incredibly dull to level so maybe don't worry so much about it.

Bridge Officers:

Tactical
Commander: Attack Pattern Omega III, Cannon: Rapid Fire II, Cannon: Rapid Fire I, Tactical Team
Lt Cmdr: Attack Pattern Omega I, Cannon: Scatter Volley I, Tactical Team
Ensign: Torpedo Spread I

The idea here is to chain Attack Pattern Omega to keep it up at all time, then chain CRF while you focus on large targets. If you're going after groups of weaker ships, couple CSV with Torp Spread to ruin their days.

Engineering
Lt Cmdr: Reverse Shield Polarity, Emergency Power to Shields II, Engineering Team I

This is your first 'keep me alive please' officer. Use as necessary.

Science
Lieutenant: Hazard Emitters II, Science Team I

Your second damage control officer. Science Team clears some debuffs and heals your shields. Hazard Emitters clears the most damaging debuff in the game (plasma fire) and heals your hull.

Note that as of the latest or next patch, Tactical, Engineering and Science Teams now no longer share a cool down, making them all viable on a build.

I hope this helps!

Finally, every character in the game should pretty much skill out like this to maximise DPS, survivability and available subsystem power. http://imgur.com/sTgEqUK