Your economy doesn't expand quickly, really. even with focusing, it's generally only a few % a year, but that really adds up over time. Earth is going to be the primary focus of your production economy for quite some time, simply because of how long it takes to build new facilities, and taking production away from earth makes it that much slower. But yeah, spamming mines/automines really are the way to go, because if anything gets in the way of your expansion it's the mineral crunch.
1. I leave off my first colony decision until I complete my geosurvey of the system. if there's minerals on mars, I colonize mars. if there's luna, I colonize there. if none, I colonize luna because they're closer. even if you don't make much money, you're growing people there and more importantly a civilian trading structure. being able to issue civvie contracts to move stuff greatly simplifies colony micro from faffing about with freighters and routing and waypoints and such to a simple 'supply automine x40' order in one place and 'demand automine x10' order in four other places. civvie ships also create infrastructure on their own when trading, which means that you no longer need to sink thousands of duranium on that to get a colony up to a usable pop.
2. the mineral crunch. Earth is highly limited. you're going to run out of at least a couple things with alarming speed. There are four minerals you *must* ensure steady supplies of immediately.
1. duranium. without this, you can't build anything. as much as possible, as soon as possible
2. corundium. the second critical mineral for mines. if you get blocked on this, it makes digging out of the mineral crunch that much harder
3. sorium. no fuel, no ships, no empire. the spice must flow.
4. gallicite. critical mineral for engines. if you do any major construction, you will use mountains of this stuff. it doesn't matter how much you have in your stockpiles, you need more. especially if you're going missile-heavy. that 3 gallicite per missile doesn't seem like much, until you need to replace a couple thousand expended missiles.
3. targetted research. If you've got the mercassium, population, and income stream to spare, build more research plexes. always build more research plexes. The power of the science firehose *cannot* be overstated for all aspects of your empire, despite the usual focus on how it lets you blow things up better. Unless there's a specific hole in my tech base, or an obvious deficiency in combat power to repair, I mainline mining, construction, and research efficiency techs. if I have 100 automines, and research from 10->12 tons mining efficiency, that's like having just built 20 free automines at the cost of a few thousand racial wealth. And unless you're going full bore on everything, you have wealth to burn. And even if you don't, there's a tech that makes racial wealth better. If there's anything that can make your industry explode, it's that.
EDIT: some more notes, because I realize I'd forgotten some stuff after I hit post. Obviously, wait until earth's minerals are mostly exhausted before converting over, but mars/luna mineral surveys also tend to determine whether or not I convert regular mines to automines. if they've got good deposits, I just ship the mines there instead, because automines are pricy compared to the manned model. you want a sector command as soon as possible, because it stacks with planetary governor's bonuses. Planetary governors are awesome. They're basically free improvements for whatever they have bonuses for. Geosurvey team everything you put a colony on. They aren't as reliably good as they used to be, but you can still get lucky.