how do you conduct espionage missions

Don't bother. They are not worth the cost or effort. They are far too expensive and usually fail anyway.
 
Or even better yet, complete the Intelligence Agency wonder. Once completed, you are able to plant spies (at a cost). The spies can do much more espionage than through an embassy.

(edit: I should add that spies, once planted, tell you how many of every particular unit the enemy has - just by having planted them. No need for an additional pay out. While your advisor will say 'We have a much stronger military', it really makes a difference to see exactly how much stronger or weaker you are.)

Just a warning, though, the price involved is sometimes much too high to attempt. Even with the cost, they seem to very rarely succeed (at least for me). And with failure, most times, will come one angry AI. Fail against a militaristic civ and you have yourself a war.

On the bright side, if you ever find yourself needing that rival civ to attack you for whatever reason...
 
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