Oreo's spying guide (DA 1.6): (since I can't play the music for you, just hum the elevator version of "Girl from Ipanema", or if you happen to own it please play it as you read this)
To see the full effects of spying give the Krynn a spin. I configure them for max spying. Any race will do, but if I want to see tech stealing, and not stumble randomly on it, I pick Krynn. If I don't play as the Krynn, I don't spy on the Krynn, it's a waste of spies, because they will neutralize them and return the complement. I start making spies right away on the first turn, increasing the espionage spending slider to 25% as soon as my economy can handle it and I leave it there.
I spy on my enemies first and my friends later. For example if I am the Drengin, I spy on the Torians. If I am the Iconians, I spy on the Yor.
I don't just send in one spy, I wait, send six, and keep a couple in reserve. I put spies on their economy structures first. The structures that are vulnerable to spying have a green box around them. The structures that can't be spied on are not boxed in, like empty terrain and military structures. I always look for that Econ capital. If they are at war they will be particularly vulnerable. They will neutralize some of my spies at first, but they will have to make a choice with their resources. The AI tends to choose military and social production over spying. The AI will always choose military over spying when at war. I just keep adding spies to their worlds to deplete theirs. Note: Spies are available for reasignment after a world is conquered.
For military conquest, I neutralize farms with my spies. For influence conquest, I neutralize farms, embassies/(insert upgrade name here), and the political capital. For Tech stealing, I put my spies on labs. I've never stolen a tech from a farm or market. Neutrality learning centers seem to be harder to steal techs from also. I had 15 spies all on neutrality learning centers on Thala for at least 10 turns, the planet flipped via influence and I did not steal any techs from them.
Espionage levels:
Low - gives some basic data on the race and shows where ships are going, this is useful.
Medium - gives some more data including economic data, also useful now I can see how to hurt them.
High - gives everything I need, and don't want to know. Most importantly, I can double click their worlds. This is a convinience, since I keep at least a few spies in my pool I can click spy on a world and always see what's on it.
Advanced - more data that I don't care about, their bonuses are already in the game manual, and the rest I can figure out more or less from their upgrades and starbases. So what if I can double click the starbase I am about to blow up to see what modules were on it.
In the late game spying can become as useful as avant garde gourmet professional rabid weasel polishing, so I turn off spending and apply as necessary.
(Stop humming or turn off music.)