Early expansion is one of the keys to the game. It is not the only key to having fun with the game. When the galaxy is filled with settled planets, having more planets than anybody else makes everything else easier. When you get better, add more AI players until you get the challenge you want. Then you can try to overcome their advantages.
Being attacked in turn 286 sounds like you had plenty of time to research and build a military fleet. My practice is to research at least one military tech between each civilian tech. There are better ways to handle that, but it is a good start. I research speed techs, enough to get survey ships, and then the largest hull I can build. By that time, I have hopefully met others enough to get an idea of what weapons they are using. That decides what defenses I will research. Choice of attack seems less obvious to me, so I often pick missiles because I like them. I max out one weapon and put a little research into the others.
By that time, I am way behind on military strength, so I get very busy putting out the newest and biggest I can. Then I start looking at what I can do for support ships and other things to be found in the tech tree. I recommend exploring and trying different combinations. It is the only true way to learn what will work for you. Remember to research the techs that get you additional capacity. Nothing makes a ship better like more room to put stuff in.
I don't understand what troubles you having with the ship designer. If it is problems designing and moving parts for good looks, check the forums for a couple videos out there. Still, trial and error is the best way to learn. I recommend having a kid help you. Their view of the shapes is very different, and together we made some wonderfully silly things. 
If your issue is deciding what equipment to use, then I can only give you my present strategy, which is still changing as I learn the game.
Add engines. No good building them if they can't get anywhere in time. Add weapons. I often pick one weapon type per ship. How many weapons is enough? There is never enough. That is the problem. Isn't it? Take your best guess. I like to end up with more weapons than defense. Add defenses. I like to either concentrate on a defense type according to my current opponent, or, if I am not fighting anyone at the moment, spread the defense across two types per ship. I find I don't need as much range as I used to think. Sensors are always taken care of by a nearby sensor barge. (If you don't know what that is, look around the forum. It caused some interesting discussions!) I have to repeat: this is just what I do and I am not the best and I will be changing as I learn.
Last but not least, research logistics techs. Nothing helps a ship like more ships. 
Enjoy the game. Be sure to make plenty of mistakes. You may try things we never thought of and come back to teach us.
