Game begins, you have a frigate producer, two con ships, and a terran planet...
-immediately upgrade your civilian infrastructure for more Credit income.
-get your crystal mine started, then queue your metal mines up behind it.
-build 2-4 scouts, depending on the map. Set them to explore the moment they come out.
-build no less then 3 light frigates (cobalts or the equivalent) and throw them at a asteroid as soon as all 3 are out.
-build a colony ship
-attack, clear, and colonize a nearby asteroid.
-wait, buy crystal if you have to, and build a capital ship factory, and get your capital ship out.
-by this point, you should have begun expanding. Throw your capital ship (yes, alone) at a lowly defended planet. If the planet has few enough ships, the capital can deal with it, send the frigates to help when they are done. Use your colony ship to colonize both.
After this point, its pretty much varied depending on situation...
-save up, buy a pair of CivResearch stations.
-research volcanic and ice colonization as needed.
-Anytime you colonize a planet, immediately purchase at least one Civilian upgrade in order to get it out of the red. Two is best if it can support that. Make sure you save at least 4 logistic slots in EVERY planet you colonize for one trade station.
For military research, Armor > Shield > weapons. Only expand to higher tiers when you have exhausted all the research in your current tier, have just colonized a desert planet and have additional resources, or need something specific.
when you hit the fleet cap, and you will, don't be afraid to take the first fleet upgrade. The second one should be taken when you begin an actual war with another empire, with the remaining ones coming as you need them.
For long Research Queues - you know what I'm talking about, when you have 10 or so different researches all lined up. Unless you have a massive flood of money, cancel a few of them, and use those resources immediately in your empire.
Trade: Trade should be gotten as soon as possible. This means the Vasari should get a number of CivResearch stations out early. Once you have this researched, take a moment to build a Trade station in every planet you can, making sure you have as long of a 'line' as you can. In order to maximize the length of this line, it may be imperative that you neglect to build a trade station on certain planets.
Tactical slots: PJI (it sucks, but it can be useful) when it becomes feasable, Repair bay (atleast one) when you have the resources, and a whole lot of hangars once they are teched. Put the PJI near the lane headed towards your empire if you are having problems with JANs (Jumpers After Nuking) or near the lane in if you are having issues with massive fleets coming in, doing damage, and retreating to safety without taking many cassualties. The repair bay should be placed where your ships intend to be while defending the planet, and the hangars should be bunched up. If you have enough TecPoints left, put a turret or two down next to the hangar cluster, for added defense.
Split fights and bombers from static defenses 50/50.
Cap ships should have all Fighters UNLESS its a carrier, then 50/50 is fine. This is so your Cap ships have their own, natural anti-fighter/bomber defense. I personally run my fleets with all fighters, I figure my capital ships churn out enough damage as they are.
The best military advice I can give anyone is Bottlenecks! Find those planets, we all know which ones they are, that are in the most direct route from their empire, to yours. Colonize them, and stick a good fleet there. This helps defend from unexpected attacks. Having 4 lanes between your empire and your rival's is good and all, but having only two will be much easier to defend, and will require significantly less resources in the long run. If your enemy claims a bottleneck, push and try to take it, if its feasible.
At least one capital ship should be in every major defense fleet. Two is better. Which capital ships you build depends on your race. Play single player to get familiar with every Cap, what it can do, what it can't do. Where its strongest guns are, where its got weak or a lack of guns. Your first should be a strong Battlecruiser (Kol, Radiance, Desolater) Something that packs a punch. Your second should be one with good supporting abilities (Carriers, high grade ability caps, Ect)
For ship abilities, your first capital should focus damage, your second, survival, from there, get what works.
Carriers first aren't good, for it makes it difficult to colonize planets with flak frigates.. which is a great number of them.
Only build Culture structures if your enemy does, save the resources otherwise - but leave a few logistic slots here and there just in case. Especially on border worlds. The best defense against Culture spam is a fleet of capital ships in orbit.
Fianlly, Don't neglect cruisers and Frigates. 16 capital ships as an armada is good and all, but when the enemy has 100 LRMS, and thats all you've got, you get to watch the miracle of massed focus fire.