Construction sites in space, mining the Moon, plus discovering new ways to propel the spacecraft. Also need to worry about shielding the astronauts from high energy particles, etc. Our early space probes are already on their way. Pioneers 10 and 11, and Voyagers 1 and 2 are continuing on and will continue on until they run into something. Just in case, all 4 of them carry plaques and recordings to greet any curious space alien.
Chemical reaction goes on, heats up gas, channelled backwards so that rocket goes forward.
Also there are courses on space and planet exploration offered here. Nuclear fuel would give more bang for the buck. Less mass needed to generate more energy. A part of the mass will get converted into energy. This idea dates back to the 1950s, both the USA and USSR investigated it.