I only have a few suggestions for you, and I warn you that my style of play and strategy may not suit you. But here goes:
First, decide what kind of war you want to fight. I like tech supremacy, but if the Drengin hordes are coming over the border you may have to pump up production of inferior ships and fight a war of attrition, at least until you can build up and take the war to him. As a last resort, try to preserve your ships to kill his transports... Always try to wage war in his space, if you can. If nothing else, take a planet and offer it back to him for peace - that might buy you some breathing room.
Second, I like to have a good logistics level - around 8 small or 5 medium ships in a fleet seems to give me a numbers advantage through the mid-game. Remember, it's not just how powerful your shots are but how many of them you can get off before he fires back. To limit the damage you take you ideally want to kill his ships in the first round and that's easier to do if you have a lot of ships shooting.
Third, I like fast movement. I'll fit two engines onto every warship even if it means lightening weapons and defenses. This lets me get my ships from the shipyard to the front quicker, and lets me strike against multiple targets in one turn. It also allows you to switch from offensive to defensive if you need to. A single fast fleet can replace several slower ones - assuming you can win the battles and keep bringing up replacements. A fast constructor or three lets you snap up his resources before others can get there, assuming that you are able to kill the starbases.
Fourth, tech. Tech, tech, tech, tech and, when in doubt, tech, with a side-order of tech and tech for dessert. You need a good weapon and defense, good engines, good miniaturization, good logistics, and at least a small or medium hull to put them in, so research can't wait to the last minute.
Fifth, production (after you research the techs - did I mention that tech was important?). Can you out-build your opponent if you have to? Because better ships are nice, but more of them is lovely; as Stalin supposedly said, 'Quantity has a quality all its own.' I almost never put a ship in orbit (or at a base) unless it is damaged or there is a threat from a nearby unsupported enemy transport. Garrison ships are a waste of resources; if they ain't fightin', you don't need 'em, and if you have 'em, you should be usin' 'em. A fast fleet can guard several systems just by being able to move rapidly to intercept invading forces, and it is cheaper than lots of garrison hulls. And lots of fleets can smother an enemy; every ship you kill is his investment of production gone, so make sure you are killing his and not losing yours.
Sixth, build up an offensive potential with fast (really fast - 2, 3 or 4 engines) transports. Your fleets exist to peel away planetary defenses; your transports are the ammunition you expend to win the war. Every planet you take away reduces his ability to fight and will (over time) add to yours. I can usually keep an offensive going until I run out of transports, so having a 'stockpile' lets me flatten an opponent in a single campaign. The key to victory is real estate, IE turning his into yours. Transports that can strike from 11-17 hexes away make GREAT real-estate agents. Who cares if his fleets cruise around, if you are taking his planets out from under him?
Seventh, do your homework. If he has beams, you need shields, and if he has shields you need missiles and so forth. (I wouldn't use kinetics under the present setup, but your game/your decisions). Try not to let the enemy have a weapon you don't defend against - that costs you lots of damage (and ships destroyed), and shooting against a defense he has prepared to receive you is a waste of time and combat rounds. Obviously this is easier to do when you have time to prepare (IE you start the war, not him) but it is really important that you not give the enemy what amounts to free shots.
Eighth and last: Nobody wins a long war. Ideally you want to crush his combat power by zapping his fleets and taking his planets, and you want to do that quickly so that he can't build more warships to hurt you with. In battle, use a weapon he can't defend against, and to win the war take his planets.