My favourite time to photograph is just after some rain has fallen, generally at night or in blue hour. I like the way that artificial light from shops, streetlights and cars reflects off the streets and roads. You can use it to really isolate and emphasise a subject.
Precipitation in general can give some interesting effects, mist can act like a filter and depending on the light can luminesce and glow. Be careful when using flash in the Mist or Fog as it can reflect and dominate the exposure. Snow falling adds interest to a scene and when on the ground will blank out features and unify a scene.
I do occasionally go out in the rain but no camera is fully waterproof unless you use a housing. If your camera is small enough to use single handed then an umbrella works fine in light rain. Remember you can add flash to a scene to give even greater reflections, but if you use this at night be careful not to startle people. Remember to keep the ISO up I generally start at 3200 and move from there. I shoot with prime lenses 99% of the time and that give you the extra stops you need at night I try to shoot around F4 but will open out to F2.8 or 2 if required.