1) What sort of path do I need to be on to be having a "good" first game? Are there any milestones I should aim to hit? I can tell from here that I'll want to have off-planet mining up and running before Earth runs short on some minerals, but I have no idea what else I should be striving for. In particular, I'm thinking of economic/technological growth rates.
There is no goal except what you make up for yourself and there is no score. However, there are few basic things that are the same in every game:
First, mineral production - you need to be able to mine all 11 minerals in such amounts that your stockpile grows instead of shrinks, within reason of course, so it has to grow as your empire grows. First step here is to set up mining colonies on planets/moons/comets/asteroids so that you have sufficient amounts of all minerals coming in outside of Earth. Whether you do this via mining ships, auto-mines or manned mines, and whether you transport them via freighters or mass drivers - or some combination of all - is immaterial, though it's a good idea to experiment with all of them.
Second, fuel production - as your navy grows, you'll consume more and more fuel and Earth will run out of Sorium at some point. Again, I would recommend to set up both manned refineries somewhere where you can mine/ship Sorium easily, as well as fuel harvesters are suitable gas giants, plus tanker(s) to move that fuel back to colonies.
Third, wealth production - the third classic "crunch" hitting a growing empire is lack of wealth. So get your civilian shipping lines up & running, start colonies to increase your tax base and promote trade, and build financial centres to absorb excess population and make money. Especially with a conventional start, it's easy to lull yourself into thinking that your wealth stockpile will last forever - it will not.
Fourth, terraform stuff - run a couple of successful terraforming projects to get the hang of the atmosphere calculations and what sort of industrial effort it requires. Luna and Mars are the obvious choices in Sol - though I personally love the idea of semi-subsurface Luna City under massive domes that I hardly ever terraform Luna - and you will find plenty of solid choices outside of Sol as you explore. Colony cost 2 planets can be terraformed with only facilities since infra is cheap, but higher CC planets will require terraforming ships/bases. Again, experiment using both to learn.
Fifth, fight a battle - sooner or later you will run into spoilers if you have any turned on, or an NPR, if you allow the game to generate them or even started with 1 or more. You need to build a navy before you need it, so start fairly early - crappy ships are better than no ships - and keep making new designs when your tech advances and build 'em and form Task Groups and Task Forces. Once more, experiment with both missiles and beam weapons, as well as fighters and FACs, to learn how they all work and what you prefer.
Sixth, take over a colony - the last big thing on the check list is to wage a ground war. This might just be a small skirmish over a tiny spoiler base, or it could be a massive campaign occupying an NPR homeworld, but you need to build transport ships and train ground units and be able to send them over without getting them all killed in the process.
Once you've done all six, you've basically tried every major aspect of Aurora in some manner, and you're well on your way to becoming a rugged veteran of EXCEL IN SPACE!
