Okay... this is the way I used to do it on PC a LONG time ago, and it's funny how the memories came back to me when I downloaded UQM... it's pretty quick. It DOES however involve spoilers of a sort, so if you don't want some of the game given away, don't read further
This is kind of long, and not as detailed as I'd like since I don't have my QSpace map... but it goes into enough detail that it shouldn't get you killed.
Takes some time to get started, and you probably won't be blowing anything up but the random probe for a while, but it's worth it in the long run when you're using a full rack of Hellbores to tear into Marauders by the early spring of 2158.
First... grab Fwiffo from Pluto and get all the minerals you can from Sol's outer planets. Your hold should be full by the time you get to Mars. Visit the moon and do the starbase thing. When upgrading, and this sounds weird... sell your ion bolt gun so you can afford to buy all the fusion thrusters your ship will hold. This is key, so that you can outrun any probes chasing you in the next phase of the plan. Don't worry about turning jets yet... your turning speed will suck, but you can navigate with patience. Now, buy as much fuel as you can.
So, now you've got all this fuel. Head to Beta Pegasi I ( 039.5 745.8 ) and scan the rainbow world there. Leave, don't use up your lander. Next stop is Alpha Pavonis VII ( 056.2 800.0 ). Energy scan, and drop your lander next to the warp pod you'll find there. Next, head toward Gamma Columbae ( 072.9 620.8 ). You should run out of fuel on the way there... and soon the Melnorme will show up and want to "help" you. Sell them the rainbow world location and buy maximum fuel, plus your first couple hits of technology. Don't worry about information yet. Now you can head for Gamma Columbae... and this is where you get to wait for a while.
On the 17th of the month, and lasting till the 20th, is the QuasiSpace portal nearby... hang out until it opens, talking to the Arilou passing by, and then once you're in QSpace open up the map and head for the top right dot. This is where the Arilou live, and they'll hook you up with a portal spawner once you give them your warp pod.
Okay, you've got the portal spawner. Now it's time to go visit rainbow worlds... don't bother landing, just scan as many as you can find.
Short list, in the safest order...
Groombridge (near top right of HSpace map)
Gamma Aquarii (Utwig space)
Beta Leporis (pretty close Gamma Aquarii)
Five of the QSpace portals on the upper right of the map form kind of an arrowhead shape pointing to the Arilou homeworld... you'll want to take the portal that is just behind the portal that makes the very tip of this "arrow" point, and the rainbow worlds are just a short HSpace trip away.
Once you've collected those three worlds, then hop back into QSpace and take the portal to the very right of the lower cluster of portals... this will drop you off near Zeta Sextantis I (( 468.1:091.6 ), which is the next rainbow world you want (relatively safe). Scan it, and then head for Alpha Centauri until you run out of fuel or get there. Either way, you will eventually run into the Melnorme again, and you can sell them the locations.
Now you're able to purchase everything up to Hellbore cannons, and your landers are super-tough.
Begin collecting minerals.
Start with Alpha Centauri III and VIII, and when your hold is full, go home and grab those turning jets... one more run (I prefer Vulpeculae, since it gets the Orz out of the way and is pretty rich) for minerals and you're able to arm yourself again. Buy storage pods as you can afford them, work your way up towards those Hellbores, and use what several credits you have left over to buy fuel from the Melnorme when needed. It's best to cut and run as much as you can until you can pay for the firepower you need.
The spawner is a handy tool for escaping from those no-win situations, too... if you need to run home quickly, the Sol portal is just above and left of the one you used to get to Zeta Sextantis.
Enjoy exploring the galaxy, and try not to get gruesomely killed, Captain.