"I just started playing, and don't know what to research first. Help!"
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This is actually a very difficult question to answer, because what you're going to research depends a lot on what your racial strategy is - you basically want to research things that support your strategy. Some thoughts:
1) Aurora is about designing systems, not just technologies, especially when it comes to weapons systems. You need to think about all the technologies needed to put together a functioning system - missing one can cripple your weapons system (you'll find this out either when you try to design the weapon, try to design the ship using the weapon, or engage the enemy with a weapon you can't get to work
). Take lasers for example. If you look at the energy weapons techs, you'll see two technologies directly related to lasers: focal length and frequency. Lets say you've researched a couple of levels of these two, and try to build a laser weapon system. You'll discover that you also need capacitor recharge rate (a power and propulsion tech) - going from 1 to 2 makes your lasers fire twice as fast. When you go to design your ship, you'll discover that you need to create a beam fire control, which requires its own set of technologies. You'll then discover that a fast (anti-missile) fire control doesn't do you much good if you don't have high speed ships or decent turret tech. The best way to figure all this out is to try to do it and make mistakes.
2) Don't forget the economy. Aurora is all about increasing efficiency, and your fundamental measure of economic efficiency is the production tech levels. The big three are factories, mining, and research, with wealth right behind them (if you ignore wealth as you advance the others, you'll eventually begin to run a deficit and your productivity will go down).
3) Two important ship-building philosophies are "Speed is life" and "Armor is life". If you're fast, and have a long sensor range, then you can control the engagement with your enemies (unless they come after your homeworld). If you're not fast enough to out run them, and they can see you, then you'd better make sure you can take the pounding they dish out.
4) "What's your missile defense strategy?" There's a good chance that NPR you encounter will have them, you need to decide if you're going to try to stay out of range, shoot them down, or absorbe their punishment with your armor. Don't forget that a hostile NPR will try to use missiles to obliterate your homeworld if it can find it.
5) Gunboats/FAC are a cheap way to get early striking power in your home system.
EXPERIENCED PLAYERS - the above was just intended as a starter. Please feel free to post your own thoughts on the matter here - I've dedicated this entire thread to the topic so that the replies are interspersed with other FAQ.