An economy specialized planet is always my first planet now. By the time I colonize my third planet for technology, the second one is half done with its economic buildings, which will support it financially. Otherwise I go broke fast.
I have to agree with Evil on this one. Wait until you have all your planets before deciding what to do with them. This is especially true for your capitals; once they are placed, they are there for good. It is excruciating to build a manufacturing capital, and find that the next planet colonized has a couple of +300 bonus tiles. As far as going broke; don't overextend yourself too soon. It takes awhile to build an economy, be patient.
how does Espionage even work? Is there a % chance for them detecting the spy, or is the spy immediately known when it's placed?
When you place an agent on an improvement, it nullifies the effect that improvement gives. (as if it weren't there at all) The longer an agent is on a planet, the higher your level of intelligence will rise for that race. Multiple agents on the same planet, or agents on other planets of the same race will speed up the process. The AI sees your agent immediately, just as you do it's agents. Whether or not it responds to nullify one of your agents depends on the value it places on the improvement, and whether or not it has an agent in reserve. Placing an agent on a capital is sure to get an almost immediate response- again, as long as it has an agent available. I have found it useful to keep a few agents in reserve, in order to counter the AI's moves. Espionage can be a very powerful tool, once you learn how to use it.
I also mined some Influence resources but I have no idea what that's doing.
Resource starbases are some of the most valuable things that you can build. Grab every resource you can find. Resource bonuses are applied across your empire, as opposed to the localized effects of the more mundane varieties. And, their values stack. Their values are pretty self explanatory; economic resources give a boost to your economy, military boosts the power of your ships, research increases the amount of research you do, and influence increases your galactic influence. You must improve them to get the maximum benefit, and in order to do that you must research certain techs in order to be able to build the modules necessary. It's a bit of a process- but the end gain is more than worth it.
I guess is a good reason to colonize everything in your territory?
Not necessarily. I sometimes skip over planets closer to home, in order to get to a more valuable one a bit farther away. Wait until the other civ builds the planet up for you, then raise your influence, and culture flip it.
I wanted to kill them all.
Sometimes, that is the best strategy of all.

Good gaming.