Once we unlock antimatter-fueled propulsion we could send a probe to the Proxima Centauri system.

If we try sending one now, even if it escapes the sun's gravity well, it would easily be passed by a future probe if antimatter propulsion is invented within the next century or so.

Once the probe reaches the system, it would take a little over a decade for a radio signal to reach us. So it would need some built-in automation to run research missions because the latency is just too high for remote control from Earth. Unless some sort of quantum entanglement technology enables instantaneous communication, but that might just be scifi...