As TEC, I always try to get the Terran population upgrade, Modular Architecture, and Metal Mining upgrades first, in that order. Military-wise, getting HP and Armour research are essential, and the Missile and Autocannon researches are also worth getting. I usually don't bother with most of the Civilian or Defense trees, and on the Military Branch, the laser and beam upgrades aren't exactly stellar.

With Vasari, getting Phase Missile upgrades are essential, while Pulse Guns, Pulse Beams, and Waves aren't worth upgrading at all unless you're running a lot of Skirmishers. For their Civilian branch, the Terran upgrade is great, as are Enforced Labor and the resource bonus research.

I think there is only one technology that I have researched in every game I have ever played without fail, regardless of my race, strategy, and start position.  That is the repair bay.  It is the tech to always research.

Aside from that (on the military side of things) you have the usual "vital" technologies regarding unit prototypes.  You need a balanced fleet, and usually that means LRF, carriers, flaks, and probably a support cruiser.  I personally am a huge fan of antimatter upgrades, but keep in mind I'm also love my capital ships, so I put those upgrades to good use.

For useless technologies on the military side, I'd have to say the Vasari technology that increases capital ship bombard range is probably the top candidate.  It's ridiculously expensive for something that has virtually no effect.  Not worth your time or money.  The capital ship beam weapons are another dubious research; at least they do something, however.

Now, the civic side of the tech tree doesn't have very many "must have" technologies.  I think ice and volcanic colonization are probably the most important in the long-run so you can claim those planets both as tactical staging points and for their resources.  As Swordsalmon already mentioned, the population upgrades are a great investment and the low-level extractor upgrades are well worth your time.

Now, I could probably rant all day about weak civic technologies, but let me address an absolutely worthless one: pirate xenophobia.  This ability reduces the strength of a pirate raid against you by about 10 or 20%.  An experienced player probably already knows what I'm about to say, but I'll say it anyways.  This pirate raid is still strong enough that I have to go back and fend it off.  If I go back and fend it off, then this upgrade has effectively given me 10 or 20% less experience for doing so.  In other words, this ugprade is actually a bad thing to have.  It's one thing to be a double-edged technology (TEC insurgency can feed the enemy experience in the long-run, but at least it has positive side effects), but this tech seems to literally be a detriment.

Well, they said they would change the pirates in Diplomacy, so maybe Pirate Xenophobia will be relevent again after the expansion. But for now, ya it is a bad tech.

Almost anything that is level 1 research is worth the resources, as they are cheap and many of them are actually quite good. I usually get everything that gives access to a new structure or unit that requires less than 3 labs in every game, unless the home worlds are next to eachother.

Things I never research, besides pirate xenophobia.

1. Culture upgrades that don't give a boost to you ships (except the advent tech that allows you to see other gravity wells)

2. Vasari scrap upgrades.

3. Extra drone anima for advent hangars.

4. TEC cluster warheads (though they said they fixed them so they may be worth it now).

Those actually are pretty good.  Certainly not early-game technologies (cept the Advent one since it's available with 2-labs) but definitely worthwhile.

Situational but worthwhile if you do put them to use.

Not worthless, persay, but definitely overpriced for what you get.

As a player of vassari I've noticed that I don't need the extractor upgrades.  I'm always credit starved and just fine with metal and crystal.  BTW, while it's a late to get tech I always grab improved fleet communications from the military side.

actually extractor upgrades are the most important econ advancement on the vasari civic tree because they ahve to wait for 4 labs to get trade ports up. you can get 4 upgrades for extractors (both metal and crystal) with only 3 labs. If you run short on credits you can sell the metal or crystal at the black market.

_|~Uber

This is normal for Vasari, but because the black market allows you to turn resource income into credits, the best way for a Vasari to increase credit output is actually to buy these extractor upgrades and then sell the proceeds to the black market.  You can also get enslaved labour if you're really low on credits to reduce the credit cost of your ships.

My quick list of techs that I find less than useful:

TEC:
    Combat:
        Capital turn-rate
        Beam damage
        Missile Range
        Cielo (Without researched ability)
    Defense:
        Gauss RoF/Range
    Civilian:
        Refinery
        Ice Pop
        Trade ship armor
        Cap training level
Advent:
    Combat:
        Beam range
        Cap training level
        Scout Kamikaze
    Defense:
        Hangar strikecraft
    Civilian:
        Multi constructor building
        Antimatter constructor/improvement
        Culture resistance
        Seeker vision upon death
        Resource Focus
        Bounty on enemy for killed tradeships
Vasari:
    Combat:
        Cap training level
        Overseer researched abilities
        Pulse Gun
        Pulse Beam
        Bombing Range
        Wave Cannon
        Armor boost+Mass decrease
    Defense:
        Grav mines
    Civilian:
        Salvage from dead enemy cap
        Refinery
        Nanite Jammer/improvement
        Bounty gained increase
        Pirate raid strength decrease
        Ice pop
        Capital culture suppression

Well, if you are using them, the guass turret RoF upgrade really isn't all that bad...  In fact, it does increase the DPS by a sizeable margin...

Also, the Nanite Jammer ought to have uses.  I've personally never seen it as I can never afford to bring it out until I'm pretty much guaranteed a win, but it seems like it would have a dramatic enough effect to be well worth it...

The problem with gauss turret range and RoF isn't their power, but rather that people don't normally use gauss turrets

I don't see your aversion to the Vasari pulse gun.  It benefits the skirmisher and starbase and is cheap to research.  If you're using skirmishers (it does happen, for instance against carrier massers) this is a great one to have.

The problem with Pulse gun research is that is only affects Skirmishers and Starbases. If you need many Skirmishers, then it is a great technology, but otherwise, just seems a bit of a waste of resources.

I do have to take back my comment about Vasari pulse beams not being worth researching. If you're using Kortuls (And why wouldn't a Vasari player now?), tghen the beam research makes Power Surge even more powerful.

Well, of course if you're not using the unit-type the ugprade isn't worth researching.  That has nothing to do with the upgrade being bad or something to avoid.  If you are using skirmishers (and yes, I have done it quite a bit in multiplayer) then this upgrade is a shoe-in due to its low cost.

and your point is?

Do not research techs that you are not going to use.

mine are

TEC
     Advanced Maneuvering
     Heavy Fallout
     Civilian Ship Safety Act

Advent
     Basic Crew Training
     Communal Labor
     Martyrdom
     Karmic Retribution

Vasari
      Mobile Phase Detection
      Jump Degradation
      Pinpoint Bombardment
      Grav mines
      Raider Xenophobia
      Expert Bounty Hunting

everything else I can see being useful in some manner, these either have no effect or is a bit redundant