Here is my working list of tips for new players.  I've only been playing (this version of) the game for a week so I'm sure there is stuff I'm missing:

1. Customize your civilization to match your play style.  You can get extra points to spend on good traits like +10% manufacturing by taking penalties on less useful stuff like deceptive or intimidating.  A civilization with Spore and using Legendary ships gets you a lot of level 5 planets, if you assign leaders to them.  (Spore gives you a charge to turn a dead world into a class 5 world every time you get a culture point).  You can change the starting ships you get (for example getting a constructor instead of a colony ship can be useful when you play Yor so you can find durantium, which is needed to create more pops).
2. Look carefully at each leader you can recruit - some of them have really useful powers such as providing a culture point, or being able to steal tech when assigned as a diplomat to a foreign civilization.  Some really suck as governors, too, those scarred bastards.  Use the Executive Order to invite new ones when they blow.
3. Trading: When you meet a new civ you can talk to, sell "open borders" for 2 credits/turn, it gives you a little money and reduces political friction.  Similarly, although the AI is parsimonious on trades, you can always tech-whore.  For example, I like to research the first couple trade techs, and then sell them both to each and every cib on the same turn for a bunch of monthly credits or techs.  Civs will *occasionally* offer you really good trades, so don't be too quick to reject them even though the last 20 times you were contacted it was someone asking you to go to war with the Drengin for 73 credits.
4. Red suns are more likely to have Durantium (?)  (especially critical to know if you play Yor)
5. The most important improvements to make on your homeworld first are Entertainment Districts, as your approval is a major limiter to multiple outputs.  Max approval seems to be 96% but being above isn't bad because there are events that are worth taking that will lower it temporarily.
6. Maximize survey ships.  Keep them surveying only derelict shipyards (yellow icon) in order to get free ships, unless there is nothing else to do.  Similarly, throwing a survey module in a fighting ship allows it to earn experience and level up with nice upgrades (it is especially nice to have one of each of the three weapon types plus a survey module in the ship so you can utilize your extra strategic resources more evenly, or maybe missile+beam, and using Durantium for hitpoints)
7. Hover over the happiness of an individual worker to determine WHY they are happy or unhappy, as it's often things you can take care of without building structures (for example camping a single crappy ship in orbit gets rid of the -10% approval malus, which is a good use for the free ships you pick up by researching derelict shipyards).
8. The minimum number of turns to complete a research project appears to be 2, so if you hurry towards a specific tech you will have to come back and complete techs that SHOULD cost you 1 turn to research at 2 turns.  So it's sometimes best to finish off those early techs early on.
9. The minimum number of turns to complete a ship is 1 turn, though left-over manufacturing power seems to carry over to the next turn.  So if you are trying to crank out a fleet of 10 attack ships, you might as well make them JUST as expensive as you can to be completed in 1 turn, but not two, to get the strongest fleet you can in the same number of turns.
10. When you are sending ships far away, they often choose a path through areas that haven't been surveyed and might contain pirates or monsters.  For constructors and colony ships in particular, manually route them through territory that you have previously found to be safe, unless you are feeling adventurous.  By the same token, you can often reveal more sectors by choosing a route that has the exact same number of moves but is going a different way.
11. When exploring the edge of known space with a weak ship, move only half your turns forward at any given time so you can run away like a little bitch if you find a monster or a pirate
12. Not all level 1 planets are the same; look at the values they actually provide you before colonizing them (the AI certainly does)
13. Assigning a leader to be the minister of technology gives you +1 choices (4 instead of 5) in the technology tree, as well as a +% bonus based on their intelligence.  Also be aware that when choosing a technology to research, you can change the focus of the 4/5 choices presented to you at an additional +10% research cost by clicking on the wheel with the four choices in the upper right hand corner of the technology selection screen.
14. If you find a juicy planet that requires a technology you can get your hands on (e.g. a water world) get a colony ship moving towards it while you are researching it so you can colonize as soon as the research finishes.  Also note that if you do find such a juicy world, it's worth the 10% research penalty to select the green planet on the tech focus wheel to see if you can start researching it right away
15. Core planets should obviously be made only on huge, beautiful planets.  They suck resources from whatever is nearest, so if you create one it needs to be near stuff, and it may also take resources from your main planet.  Don't turn many planets into Core worlds unless:  (1) there are a lot of smaller level 1/2 feeder planets nearby that you can colonize. (2) there are a bunch of asteroids nearby you can mine. (3) they are far away from your other planets.  (4) you are willing to send a bunch of supply ships to it (and even then it's fairly untenable unless you have other planets nearby).
16. The most effective weapon to research first seems to be missiles, since you can barrage enemies from a distance without engaging.  You don't seem to get any experience for this, but you can wear down a pirate shipyard with a stack of tiny missile ships so you don't take any losses.  The best bang for the buck early game seems to be offense-only ships without any shields, point defense, or armor.
17. Research miniaturization when you can, and as soon as you do, re-design your ships to get that extra weapon slot installed.  A three-slot tiny missile ship is pretty righteous, especially in a stack.
18. You can bribe your planetary governors with +10% loyalty through the leaders screen (for example +10% loyalty for one harmony crystal)
19. You can change the focus of individual citizens to focus differently, for example on research.  This can give you extra approval, extra research, etc.  Especially useful when your planets have nothing left to manufacture and you don't need to keep cranking out useless stuff.  Click on the planet to see your pops, then click on them in the upper right-hand corner.
20. When you don't really need to build ships, send them out on Treasure Hunts.  The money is quite good, though there is a chance they'll get blown up, as they actually create a ship that you don't control and likes to wander around without regard for danger.
21. War is a cat-and-mouse game of colony stealing, and the enemy will send a bunch of single ships to take our your colonies, so have several fleets of a handful of fast ships near your planetary clusters so that you can pick them off and take their colonies the same way.
22. You can add colony upgrades to colonies only.  Click on a non-core world and click on the planet-looking icon at the lower-right side of the planet in the lower-left corner to see what you can add.  This is especially a good idea if you are about to turn that colony into a core world, as you can get a good (but randomly placed) improvement on it like a planetary shield or something that gives you a +3 manufacturing adjacency bonus.  Better sensors are nice when you are building close to enemies too.
23. When you find a minor civilization (and have researched universal translator) go talk to them by clicking on their planet and then the "speak to" icon (that looks like Saturn) upper right to their world.  When they first meet you they will assign you a quest.  When you complete it, you can then get them to give you something good, like a rare resource.
24. Be careful about deploying too many starbases early in the game, they really suck up your money
25. Midgame or later, economic starbases near your core planets can be quite worthwhile
26. Some of the ships that you can assign a leader to command once you reach the Minister of Defense have useful bonuses such as +20% manufacturing or +30% research if you camp them near a core world and are worth getting for that reason alone.  Be aware what special ships you are choosing (if creating a custom civ) or getting (if you chose a standard template).  It's worth rushing to the Minister of Defense tech for just this reason (plus the good fighting ships)
27. All attacks do a minimum of 1 damage, so if you are fighting a ship that has 1 point in missiles and you have 3 in point defense, you are still going to take maximum damage (1 point)  from their one piddly bottle rocket.
28. When trading with another civ, check to see if you have any resources (durantium, etc.) that they have zero of.  They will assign more value to getting one of something they don't have a mine for than otherwise, usually worth 2 credits/month.  Check back later, they may have used the one and be back to zero