I just finished the game with the Institute (haven't done the endings for any of the other factions, so no spoilers for those please) and even though I ended up being the new director I still have no idea what their actual goals are? I get that they believe they're the future of humanity, but what exactly that means isn't clear. They intend to stay underground with their limitless nuclear power, ok, but why then create all these Gen 3 synths and infiltrate communities on the surface? What are they trying to achieve?
They kidnap people from the surface to use in their experiments. The synths on the surface serve two purposes - to replace the humans they kidnap for those experiments so that no one notices, and to spy, control or keep tabs on the surface and update them on what's happening with the world above. One of the latter, you'll find out about if you've taken a particular quest for Ayo or finished the game with a non-Institute ending and then return to Diamond City.
This is not counting the Coursers, who also look human, but have no emotions. Those are designed to simply retrieve escaped synths who've gone rogue. They're basically Terminators in terms of lore and design.
I'm assuming the reason why Gen 1s and Gen 2s have been deployed to the surface even though the Institute's modus operandi is to keep anonymous, is that they're simply rarely deployed, and the only people who could've ever threatened them enough for the Institute to actively send those cannon fodder androids is the Railroad, who people barely have any contact with anyways. This is why the Institute has been able to stay undetected since the bombs fell.
For most surface operations they use hired agents like Kellogg or SRB Coursers like Armitage from Fallout 3. By the time of the events of Fallout 4, their Coursers are stretched thin due to increasing numbers of escaped synths, so they're forced to start using lower Gens more often, which is why they're a common enemy during the game.
And more importantly, how do I find this out in the game? Because none of the department heads or even Father told me what their long term vision is (or maybe I missed it?)
A massive amount of lore in Fallout 4 can be found in dialogues between NPCs. Overheard information can be very enlightening. It's a very annoying method of storytelling, I know. Some of these can also be found in terminals, and new dialogue unlocked with several NPCs by progressing through the Institute main questline, but you might skip some of them if you simply didn't talk to them between the main quests.
