Morrus, correct me if I'm wrong and please do, but it seems to me you think that because we can't do something now we can't do it ever. If the fear of dying off as a species is palpable enough, we will gladly bankrupt the world if it means we have a chance of survival.
Granted, everything we say here is speculation. If contact had been made in the 20's/30's before WWII, before the Cold War, we might be more trusting and more curious. Such a discovery, however, would have come right after WWI and during the Great Depression so we may not be as innocent as I think.
Umbran, I think, made the assumption that they are humanoid but what if they are a hive-mind type of aliens? Have you ever tried to have a conversation with an ant colony? Note, a drunken monologue doesn't count. You get nothing but a swath of red bumps and burning skin from that.
I agree my little joke about planet-busting weaponry is fanciful and silly, but you can't underestimate our ingenuity concerning destructive devices. We will find ways to kill things if need be.*
I do think that communicating for decades is the first thing that will be done. But while this happens, I'm quite positive that while we try to communicate and understand each other, running parallel to this endeavor will be individual militaries or global collaborations commissioning research into information gathering and weapons technology.
If such research concludes that it isn't feasible to attack them, either because of fear of retaliation or as you posit that we don't have the resources, I assume that we will then develop planetary defense systems.
I think, overall, if a military exchange isn't likely, that it will be a positive thing for the world as a whole. Maybe I'm a bit too optimistic but I think such a discovery would help develop a collective human identity, eroding a lot of the social and economic issues we face today. Of course, it could easily go into a different direction. People using the discovery of another species as an excuse to further their own agendas.
As someone pointed out, once technology reaches a certain point we can create a waystation between Earth and Mars when the bodies are closest. Assuming they have a desire to meet us in person this could be the start of face to face diplomacy before we start sending stuff to each other's homeworld.
* Concerning the invention of the flamethrower: "That man over there, I want to set him on fire but he's just too damned far away."