It's ironic that Neil Armstrong dies just as I am trying to organize a student group dedicated to space exploration.
Sadly, I don't think there is anything really worthwhile we can do with manned space exploration for several decades now.
Mars is the obvious next goal, but the technical problems are no joke -- especially radiation shielding. Also, we really need to be sure there is no life on Mars before we send people there; I'm less worried about bringing Martian bugs back here than I am about contaminating Mars with Earth bugs. It will probably take at least 30 years to test the most likely niches for life, especially underground aquifers, where there will be liquid water and protection from radiation, and the neighborhood of Olympus Mons, where there has been recent geological activity and where there may still be geothermal energy that could be tapped.
The idea of visiting a near-Earth asteroid sounds good, but the reality is that they are moving fast when they pass by, so there would be no room for delays, for example due to weather. The low gravity of an asteroid would mean astronauts would not really be able to walk on the asteroid; most likely, the best way to get around would be to treat it like a shuttle EVA and rocket everywhere.
As for the Moon, there's nothing much there. There may be a little ice in the perpetual shadow of craters near the poles, but we should not waste that until we know what our plans for the Moon will be. Even when we decide to exploit that resource, it will take many tons of industrial equipment to do so.
In the meantime, we should not be standing still. We are still in the golden age of planetary exploration, with fantastic results being returned from robotic probes to all the planets -- including, in about 2 years, Pluto. When we are ready to go to Mars, we will know whether the planet has life, and if so how to protect it. We will know where we are going, what it's like, what we will do there, and how we will do it.
In other words, when we are ready to send manned missions again into deep space, it won't be as explorers, but as settlers.