juanfgs, I'm not especially brilliant at the combat side, or any of it really, but I can offer a couple of tips.
1. This game is unlike any other "4X" style in my experience, in that you will grow much faster if you don't
rush out and grab planets in a huge rush. Explore, work out which planets you will be developing to high level, and focus entirely on just building the necessary support planets you need. Colonizing too many marginal outposts will leave you dragging, short of resources, while the AI empires outgrow you.
2. For the first game or three, I highly recommend setting starting conditions so an AI or two are on your team. This will give you support, as well as additional knowledge about the galaxy. You can also, I suppose, note how they are developing, and learn from their example (I admit, I never really did that).
3. When building a first capital ship, skip the smallest starting design - you'll want to build a heavy carrier fleet, and then you can be confident of having equal or superior firepower to the AI in the early running. Definitely use a mix of ships in the fleet, to handle most circumstances. I tend to pick racial advantages that ensure my starting picks aren't wasted on the little ones. Of course, with the numbers you are quoting, you're already well past this stage.
(I'm not talking about custom ship design here, since that's a huge, interesting topic in itself)
Regarding early expansion, I'm doing my best to level up my first world as fast as possible. However it's still not clear to me which criteria to use to pick my secondary developed worlds, I guess I should choose one with plenty space (to develop and build stuff).
I must clarify that I've managed to build some big ships (1.6 million in cost) using dry dock. In my latest game I also found out about hulls, which allow me to reduce the cost but using ore instead of ludicrous amounts of money (sadly it was really late in game and I didn't develop miners). I still can't find a way to get enough money fast to be able to outproduce the AI.
I've tried to colonize worlds with money resources, but they don't seem to make a dramatic impact in my economy. In my latest game I achieved by middle game a steady income of 1.6 million but by that time the hoonan empire started attacking my worlds.
Also design-wise, should I upgrade or scale the designs for support ships? should my main strength of my fleet come from support ships or from big capital ships?
Edit: Thanks a lot for taking time to answer!