I like to build 10 scouts immediately and send them out to find a good choke-point(A place on the map where there are only one or two planets that will allow ships to enter my area of space.) Try to get 5-7 planets inside this choke-point if you can, but claim them quickly so you can lay down defenses heavily there.
Economy is important in the early game too so try not to research past the first level of fleet supply until you really have to. Trade ports are an excellent investment if you get them soon enough and are willing to put at least one on most of your planets.
Once you have your economy going and have a decent defense at your chokepoint, it's pretty much over vs the AI. They usually won't throw everything at you at once, so unless you're playing against multiple AI and they all jump in at once(Rare, but it's happened to me...) your defenses should hold up.
At this point, just build your fleets and try to take as much territory as possible from your enemy.
New player mistakes:
When you take a new planet, the very first thing you need to do is max out its Population Infrastructure(At minimum, buy 1 rank for asteroids and 2 for normal planets.) otherwise they actually cost you credits to own instead of giving them to you.
Make sure you build your metal/crystal Extractors if your capitol ship didn't do it for you when you colonized. (The TEC Akkan gives you 1 free Extractor for each level of colonize you take. The other races' colony capitol ships have different abilities, but I forget the details...)
The more capitol ships you keep in one gravity well, the less experience they gain per kill since it is divided among them all equally. Titans divide it with capitols too I think.
Mines: Personally, I don't use them anywhere but my choke point. If you do use them, make sure you place them manually at the points where ships are likely to jump in.
If your enemy is using culture, you must start researching and building your broadcast towers before it reaches you. (You will know they are using it because the phase-lanes will be the color their faction is using.) It is easier to spread your culture into neutral space than it is to push back enemy culture. Placing a few capitol ships in the gravity well of a planet that has enemy culture going towards it will slow its advance, or even push it back out if you have enough. Also, you gain bonuses depending on race for fighting in allied culture. If enemy culture persists around one of your planets for too long, your people on that world will rebel, and the planet becomes neutral, and will be un-colonizeable to you until you push the enemy's culture back out again.
That said, it is somewhat of a game of finesse trying to use culture offensively, since you have to balance you income between culture upgrades and combat research/ships/replacement defence structures. It's worth it if you can pull it off, but don't be surprised if it doesn't work the first few times.
Finally, you won't just get good overnight. Everybody plays the game differently. There is no right or wrong way to go about it as long as you are not neglecting to build ships. Try the different ships and researches and see what works for you best.