Early gameFor the colonizing part, use your flagship to clear planets from pirates. Always build all mines on every colony. Don't build your colonies up. The only colony you should build up in the beginning is your home planet. There you should expand it's cities, so you'll get more taxes.
Take the gauss cannon, adaptive shielding ability for your flagship, to make it able to beat the pirates faster/easier. Take a bomber squadron for your flagship.
As soon as you encounter an enemy fleet, fortify your colony next to it with some gauss cannons, hangars and if your researched it, a repair station. Build 5 gauss cannons right next where the phaselane from enemy space arrives. Build a repair station right behind it. Build 4-5 hangars, with fighter to bomber ratio of 1:2.
This way the enemy will attack your gauss cannons, which will be repaired by your repair station. Use the bombers to kill enemy siege frigates and other ships, while your fighters will destroy the enemies fighters first, and after that it's bombers. This should make you able to defend all incoming attacks in the early game.
If the enemy manages to break through, either send your Kol flagship to help out or build some Cobalts to help defending the colony.
Middle gameYou should have built some groups of Cobalts (10-15 ships) or another Kol to expand faster. Colonize as long until you encounter enemy planets/fleets everywhere. After that, try to reinforce your border planets with the defense I've outlined above.
Build jump inhibitors at every border colony.
Research and build trade stations and refineries at each of your planet colonies. Build research stations at asteroids. Research and build some communication centers at your border colonies for defense against enemy culture.
Build some flak frigates for defense against enemy fighters/bombers. Use your main fleet (which should consist of your high lvl flagship, some Cobalts, 5 flak frigates and some of the other capital ships (max. one of the others, Kols as much as you can build) and attack enemy colonies with it. If you encounter lots of gauss cannons where you've jumped out, you either send your high lvl Kol as a shield (it can survive gauss cannons easily) or you fly around them (gauss cannons don't have much of a range). Bomb the planet with your capital ships, destroy it's infrastructure with your other ships.
If you encounter enemy capital ships, focus fire on it to destroy it as fast as possible. Destroy other enemy ships also with focus fire on them. (Destroy flak frigates first, as they would destroy your bombers/fighters in seconds.)
Late gameDestroy enemy AIs one at a time, so it will remove it's culture. Don't try to colonize planets which are still under enemy cultural influence. The colony will instantly flip to the enemies side.
Build frigate shipyards at congested border planets. If large enemy fleets attack you there, build some Cobalts and flak frigates there for additional defense. This should always be enough to repel enemy attack fleets if you use it with the defense I've showed you above.
Win.
Edit: Some ship building tips.
Don't build cruisers. They're a waste of money. Don't build siege frigates, use your capitol ships to bomb enemy planets. Don't build scout frigates. You can' build some of them if you've to much money in the early game and want some faster exploration but they're not useful for fighting and you don't really need to now much about the enemy. Don't build LRM. They're very weak and only useful against gauss cannons, but you should use bombers or capital ships against gauss cannons anyway.
For capital ships:
Build Kols. They're the strongest fighters and can survive quite a beating with the adaptive shielding ability. Build one or two Sova Carriers. They will give you all the fighters/bombers that you'll ever need.
Always, really always buy capital ship upgrades when you can afford it for your capital ships. Also research the tech, that allows you to upgrade capital ships to lvl 5. Capital ships not only gain more armor, hit points/restore rate, shield points/restore rate and antimatter/restore rate. They will also shoot faster, they'll do more damage and more abilities. They will also gain access to more fighters/bombers. A high lvl Kol can house 2 squadrons of bombers for example.
Btw. this is based on the current patch against the AI, don't expect it to work against further improved AIs or against human players.
Edit 2:
Also if a superior AI fleet attacks one of your colonies. Don't try to defend it, instead scuttle your jump inhibitor, if you've one there and counterattack an enemy colony. Usually the will then pull back, to defend it's colony. Don't fight there, but retreat back to your place. Repeat this as long as necessary until you either have built enough defenses or fleets to destroy the attacking fleets.
