Till you get used to the game stick to small-medium maps, larger maps are more difficult to manage for new players. I still get confused/lost on Huge and I've had the game for months now.
The auto designed ships you get in the shipyad after each new researched weapon tech are pretty good and balanced, after a few games you can try making you own ships. Keep in mind that you can choose what type of weapon/defense your new designs will focus on. If your enemy uses lots of mass driver ships set focus to armor etc.
Trading is a good way to make money and keep a good relationship with other factions. A good rule is to send 1 freighter to each factions capital in order to mantain peace till you feel ready for war. If you intend to make big money via trade send all you freighter from 1 planet to the farthest Ai planet you can reach and also place 1-2 econ starbases with some freighter bonus modules near the end of the route.
Try to relax and not try to micromanage too much, unless you are playing on very high difficuly.
Approval rating decides how much you population grows or diminishes. At 100% pop growth rate is doubled, at 76% it's 1.5 normal, at 41% it the base growth. Under 41% your population will start to decrease so avoid that. The values ar per planet not overall meaning that having planet A at 100% gives it 2*1 growth rate while planet B at same time with 41% will lose pop.
Try not to cram to many factions in a too small map or there won't be room for expansion and wars will star sooner( also it's less fun ) I recomend tiny - 1Ai, small - 2Ais ,medium 4is, Large 6Ais and huge 8Ais.
You generally can't get econ/research treaties nor alliances before the 2nd United Planets meeting. It's allways easier to ally yourself with a faction with the same alignment, you can see what alignment they are in the foreign policy tab>stats. In order to get an alliance treaty you need a relationship status 'close' : equal or better military rating(fleet size), trade routes(the more the better), preferably you borders don't touch and you don't have any of their planets near you sphere of influence, econ or research treaties between you, you can also gift them credits or ships for a quick rise in relation.
At the start of the game focus on colonising planets above 10(11,12 etc) these have a 10% aproval bonus of rplanet quality. Also avoid colonising planets with 4-5 till you have colonized all the the bigger planets you can find, they are not worth the upkeep.
A good build order for your first colonies is Recruitment Center/Starport, 2 Factories followed by econ buildings such as markets, do prioritize building on tiles that give bonuses such as labs on tiles that give % bonus to research buildings.
If there's anything you want to know just ask ill be lurking around.