Night photography together with a landscape and HDR photography is one of my favorite photography techniques.
Over the years I have mastered my night photography skills from completely ruined and completely dark photography to tack-sharp and well exposed night photography.
Here is what I've learned during the past couple of years - a decent DSLR camera is highly recommended, though I believe you could achieve fairly decent results with a point & shoot camera. You will be truly amazed by results DSRL can proceed. Don't worry, an entry-level DSRL cameras are not expensive nowadays, you can perfectly start your DSRL journey under $500. In fact, for the past 2 years, I have been shooting with Canon EOS Rebel T5.
A sturdy tripod, you could probably cheat around it, but to get a tack sharp images tripod is more than welcome. I'm using a MeFOTO Aluminum Globetrotter tripod, which can be turned into a monopod with ease (you can get one for about $250)
For learning more about Night Photography please read following articles: