Hi,

We have astronomy missions that study many stars in great detail and there are new ones proposed and being built. In fact the next big telescope is the James Webb telescope and it will be launched in 2019.

In terms of sending a spacecraft like Solar Probe to study another star up close, that's currently beyond our technology. The closest star is Alpha Centauri which is about 4.3 light-years away from Earth. That means a spacecraft traveling at the speed of light would take over 4 years to get there, and we aren't remotely close to making spacecraft go that fast. Solar probe will be the fastest spacecraft we've built and at its fastest speed it will be only going about 0.0007 times the speed of light.

-Christina