So, what do you think is the best investment of your resources? Of course each race has its strengths and weaknesses in each of these, but how do you exploit them?

I personally play the TEC or the Advent; I suck as the Vasari.

As the TEC, I love building a frieght train economy. Building a few small mobile defense fleets and constructing/researching economic things is what I spend on at the start. Usually before I have fleet Logistics 3 I'll have 3-5 colonies and will be getting over 50 creds/sec. With this income I research the expensive techs and take my fleet to the max. I build up a few defenses, then I start on researching/building the bigger fleet logistics. I always try to build as many trade ports as possible while throwing in a refinery or two (Creds are the ultimate resource: you can trade them for the others). Against the AI, as long as you can protect yourself in the beginning, you'll roll right over them. Against humans, this strategy works well against slower or more research oriented players. A good rush player will cut you down before you can get started, but if you ever can beat them back you'll easily crush them. Against someone who uses the same strategy, prepare for a long, weary, battle after massive battle. In these games, one colony can make the difference.

I would tell you about how I do the advent but I'm not very good with them.