I won my last game as Drengin.   When I saw that the hostilities were about to start, I would bribe some of the most hostile races to attack the others.   This delayed the war declarations on myself. 

I used the minor races as banks, selling them technologies.  My preferences for what I sold them was:

1- Things that would increase their trade and economy (so they have money to buy tech)

2- Things that increase their tech.  So I can trade for their tech

3- Things that make them less of a prey to other AI.  Usually older techs.  So if they trade those with other races it isn't as bad.

The AI were very sensitive to my troop movements.  The Arceans declared war on my when I moved troop transport into my fleets from one of my planets.

At one point 4 races had declared war on me.  I took out their home planet, this gave me a civilisation homeworld each time increasing my influence.  When they were bloodied, they seem more willing to trade for peace.  I had to make concessions to some races in order to reduce the number of races at war against me.  As a Drengin you are considered the agressor even if the other race declared war against you.    The other races in my game were:

Torian, Arcean, Alterian, Krynn, Terran, Thalan, Korx, Iconian.  The only major race that did not declare war on me was the Korx.  (They were the last race to go).   I played with Ascension and influence victories disabled.

I built stock markets as soon as I had the tech but was close to bankruptcy several times.  If you don't have enough of an army, the bullies come out to attack you & wipe you out.  If you do, they gang up on you.  I changed to "Neutral" midway through the game to help reduce the number of times war was getting declared on me.  Early on in the game having "Planetary Invasion" tech seems to discourage some from attacking me right away.

Early in the game, I only once declared war first against a minor race, and it was to get a homeworld.

When I first started playing Galactic Civ, I was able to win the game but was relying on big ships that could take out entire fleets because in lower graphics resolution I couldn't figure out how to make fleets work. 

Now, I use fleets.  My favorite combos are Miniaturization and logistics.  I don't research weapons so much but I trade for them.  When I have the chance, I combine the Xalanx  or Zalon mainframe with the Weapon amplifier tech.  It makes a big difference in fleet battles.  (I try to have 1 Xalanx / Zalon + 1 Weapon Amp per fleet).  The Drengin have a tech that can increase a weapons energy.  I couldn't find spore ships.  I presume it is a tech unique to the Drath.

I often borrowed to purchase troop transports.  I'd take out the an ennemy homeworld if I had a fleet strong enough to do it and invade it.

Once I messed up crippling the Altarian because I didn't put my troop transports in a fleet.   The Altarians were able to fight off my first invasion and take out the other attempts.  I was still able to negotiate a peace because I took two of their worlds out very quickly.

Sometimes, when my small fleet was outgunned in a sector and I needed to even the odds, I would buy whatever fleets or junk the other races in the area had.  Even when other races won't sell you military ships because of your army is too big, they will still sell you scouts, freighters and miners that can be converted to military use.  I conquered one planet with a Colony ship purchased from an Alien race converted to a troop transport.

Personally I consider the Korath to be one of the easiest races to play, probably because they're the one race that forces me to play aggressively rather than turtle to a tech victory... Go for spore weapons when some races don't have a military yet, choose which one(s) would likely become your worst enemy/enemies and capture their homeworld. Claim a few other planets, make peace and preferably get econ techs or cash out of the peace offer.

As JacquesGauthier mentioned, trading with minors is key (get cash, morale & econ techs and econ treaties). What I also usually do is get some of the standard econ techs from other races through trading so that I can build banks or stock markets rather than malicious bazaars that take much, much longer to build.

Cheers Noctilucus and Zarnick, much appreciated. Drath are my favourite race, so it's an effort to play aggressively instead of just building a huge economy and buying my enemies' empires!

I guess with the Drengin and Korath it's the economy side of things that I have the most trouble with - sustaining both the war machine and the research effort to keep it up to date. I imagine there's also an element of me just needing practice - I'm playing TA and haven't quite got to grips with the different manufacturing, research and econ structures. Teching up through invasions sounds like the right idea though, and while my last couple of games have been with tech trading off I think I may bring it back so bullying for techs through peace treaties can be an option too.

With the Super Dominator ability, am I right in thinking that the bonus ships only come once per turn regardless of how many wars start? I've had games where I have had multiple wars start in a turn for one reason or another, and seem to get fewer ships than I expected.