Update: It appears that this argument is, in actuality, a subset of the Fermi Paradox. I just came up with it on my own from a different direction. I'll leave it here, with a few modifications, for posterity.
SETI, the Search for Extraterrestrial Intelligence, is a neat idea, except that it's pointless. If you think about it, then it becomes pretty obvious that we're the first to actually look for intelligent life, anywhere, and we're not going to find it.
There is something called the Drake Equation that shows how likely intelligent life in the universe is, but I'm going to present the argument in a slightly different form. The basic gist of it is that if intelligent life other than humans exists in our universe, it would already be here, and we would not. It's pretty obvious when you look at it.
If faster than light travel is actually possible, then, eventually, some species with expansionist attitudes will start settling the galaxy. How long would they take to settle everywhere? Well, if you look at how far we've come technologically in the last while, then you realise that the time between "evolved for basic tool use to settle the entire galaxy" is not that long. If there were another intelligent species out there that did have faster than light capabilities, then they would have been here and settled the place long before we started bashing rocks together. If we're the first, then we'll invent the faster than light drive and we'll be everywhere before any other species gets to the "bash the rocks" phase. The odds of the very first species evolving intelligence and interstellar travel did so at nearly exactly the same time as another species is too low to consider, unless you go one step further and buy into the whole "we were seeded by an advanced civilisation and we have long-lost cousins out there" story.
If you agree that faster than light is out of the question, then the whole thing just gets slowed down a little, but not really enough to make a difference. We will eventually either make ourselves immortal or replace ourselves with intelligent self-replicating machines. We, or these machines will then be capable of interstellar travel because they aren't going to care if it takes 50,000 years to get anywhere. It's still within their 'lifetime'. In other words, eventually, some species will evolve to the point where they will eventually go everywhere in the galaxy. If it has already happened, then they would already be here. Even with transit times that exceed tens of thousands of years, the odds of them existing, but not being here yet, isn't even worth considering. If they are here, then we get into a whole other story about them not being particularly concerned about intelligent pond-scum like us. Unless, of course, they are studying us as "machine evolution" in action. But, that's pretty unlikely too.
So, by the basic argument of "they're not here and we are," it seems pretty likely that we're the first, at least in our galaxy. So, you should feel special. It may turn out that our descendants, either flesh or mechanical, may happen upon some species, flesh or mechanical, that does not have any desire to expand into the galaxy. That is a possibility, but we're still first to expand, and I suppose we'll crush them into non-existence fairly quickly. We are, after all, quite expansionist by nature, must be all that rat ancestry. If this argument seems a little harsh, you just have to realise that we're the only intelligent habitual tool users on earth because we evolved to use tools first. We won the evolutionary race to this niche. If whales or monkeys started using tools enough to compete with us, we would eat them. Any environment will only have one species in the "intelligent" niche. If you think about it, given sufficient time, the galaxy is one environment. Only one species, flesh or mechanical, will win the race. We exist, so we're in the lead, so long as we don't blow it.
You may think that the galaxy is such a large place that two intelligent species would have lots of time to evolve and begin expanding. Thus, we may run into other species out there. But, you have to factor in exponential expansion. If we settled two extra-solar planets, as we or our descendants eventually will, then those planets will eventually each settle two planets, and each of those settle two planets... Well, it doesn't take long to settle trillions of planets when they're being settled on an exponential curve. Even if it took a few million years from the first colony ship to the last - which is more than long enough to settle the entire galaxy, even without faster than light drive - you have to factor in that we've only been human for 40,000 years. A blink of an eye on the 14.5 billion year galactic time-scale. What are the odds that the first two species are evolving at the exact same time? If they evolved a million years or so before us, they would already be here. If they evolve a million years from now, we will already be there.
The Drake equation, with commonly accepted guesses for various factors, suggests that at least a couple of species exist in the "interstellar communication" stage at any given time. Thus, SETI. The problem with this is that the equation assumes interstellar travel doesn't happen, that species destroy themselves before they start. Maybe, but it doesn't seem likely. If we ignore world-destroying solar flares and the like, then it seems most likely that man-made destruction would encourage, rather than restrict, interstellar travel. After all, if we destroyed the earth, that's a good reason to start sending colony ships to other planets. It would be highly unlikely that something we did actually killed every human being, that destroyed our ability to breed. Total war, nuclear winter, even biological contamination, any imaginable man-made destruction, will still leave some people in protected places. Humanity will continue; we will colonise other planets. The same goes for other species on the same technological curve.
When you look at the reality of the situation, there are some pretty long odds on SETI finding anything out there. If we are the first to evolve past the point of bashing rocks together, then there is no one else to talk to. If we are not the first, then that species is already here and watching us, and they clearly don't want to talk. No, SETI makes the assumption that interstellar travel is not possible, and we know that's not true. Even if it takes 20,000 years to travel to the next star, we'll be there within 25,000 years, and then the next star, and the next, until there are no stars left unexplored in our galaxy.