First off, build an Akkan (TEC) or Jarrausal (Vasari) to help with expansion. As you take the planets immediately invest in development. Stick with frigates, research the Javelis (TEC) or Kanracks (Vasari) as they give you the best bang for your buck. Scout, Scout, Scout and find the choke points of the map and the best planets and head for these.

This might be the SMAC player in me, but I also prefer to get my economy rolling first over going hard on the military, because whatever inroads you make into econ in the early game will have a ripple effect later in the game, which will give you a much bigger military advantage in the middle and late game than you could have gotten with the same amount of resources in military units in the early stage. That does not necessarily mean that you neglect your military, but you merely go with the minimum necessary, and maybe a little extra. Just a thought.

Per usual in the RTS format, the decision to use early pressure versus booming is entirely dependent on how far you are from your opponent, and how many other players are in the game. I don't find the SMAC comparison apropos, for a number of reasons. First of all, there's no way to 'Population Boom' in Sins. Second, expansion even to neutral planets requires military force. Third, there's not really much in the way of 'lynchpin' technologies that really revolutionize your economy. The only really huge economic tech is the Trade port, and as Darvin points out, you need a good number of planets in a line to make that work out for you. All of these factors conspire to make economic development take a decided backseat to the growth of your fleet.

If by "processing plants" you mean Refineries then that largely depends on what planets and neutral extractors you own. Refineries have nowhere near the same income output as a decently long string of Trade Ports, but places that are good to build one are those where the refinery can reach a large number of extractors (I usually go for 12+).

As I play Vasari and Advent exclusively, I've never really found heavy use of refineries to be very helpful. As Vasari, having neutral asteroids from my scouts tend to put me in a position where I'm credit-bound, rather than crystal/metal bound. Sure, if I have a really well-positioned system with lots of good asteroids available, I'm not above using a refinery, but as often as not I wind up selling the excess resources on the black market. Advent's alternative to refineries are simply terrible. Don't use them, focus on boosting your culture and getting trade income instead, you'll get better results.

In Sins, the placement of your economic facilities is crucial. Build a long trade route, then consider stacking trade ports at a safe hub on the line where you've got logistic slots to spare. The systems where you don't want to build trade ports for fear of producing a shorter route, that's where your labs go. Ship building facilities should go toward the front, though you often want them to be a jump back from the front line, to keep them from being sniped by a surgical raid.

Getting a good balance of crystals/metal and credits is just as important as having a strong economic base. While you can play the black market to make up for a shortage in one resource or another, the exchange rate is brutal. In practice, your economy is often only as strong as your weakest resource stream.

Unless you're anticipating a rush (you shoudln't on the 32-planet random map) you should build your extractors before your labs and wait until you've scouted at least 1 jump from your homeworld before you decide to put down labs.  Money is tight early-game, so getting those extractors operational is high priority.  Generally speaking your best approach is to put down 2 civic labs early on so you can capture ice/volcanic planets (there is almost always one adjacent to your homeworld) then use the newly acquired planet's space to build your military lab. 

Here's the problem; you need two civic labs to do this.  If you already built a military lab on your homeworld, this means you need to shell out a lot of money for logistics planetary upgrades.  Most of the time you're better off delaying assailants and soloing the first planet with your capital ship to avoid this expense. 

You can build other frigate forces to simultaneously capture other planets with light levels of militia.  Asteroids have virutally no defense, and some ice/volcanics have as few as four frigates protecting them.  You can very quickly amass a respectable empire by moving quickly in multiple directions. 

Pick a corner of the gravity well to jump into, then send a few "scapegoat" units to absorb the mines.  Scouts and flak frigates work best; any survivors of the group will then have the ability to attack nearby mines clearing out even more.  Once you've punched a sufficiently large opening, send your fleet through and run it past the minefield to clear space.  The AI will rush to fight you on even ground, negating its massive investment in this minefield.  Be sure to keep your fleet on "hold ground" behavior so it doesn't go charging off into the minefield because it saw a target of opportunity in there...

To clear the mines afterwards, just inch forward with scouts and flak frigates.  Keep your fleet set to "hold ground" and have your scouts move into range to reveal and then let your flaks and maybe fighters do the rest of the work.  The only really annoying mines are Advent mines, which will home in and kill units.  On lower difficulties you can actually just send kamikaze scouts into them, and the AI will actually be losing more money on mines than you are on scouts.  On the higher difficulties, the AI's resource cheat means any attrition tactic is a losing one, so you have no choice but to move closer with extreme caution and knock out the mines one at a time.

I just skimmed most of these post so if i repeat someone else i apologize. When i play i first send a couple of scouts on auto explore. Asteroids of course are the easiest to acquire. Depending on the map size, i would go for a colonizer cap with a large or medium map and a battleship for small. Go from asteriods to planet, this would help get your economy going just a little and research what you need. The beginning of the game would either make you or break you. Everything is at a premium so be careful on what you spend. With thats said you should be set. Hoped this helped.