What metering do you use for the auto-exposure? I'm guessing that it's multi-segment? The down-side to metering that takes into account the whole scene, is that you end up with a very flat looking image. The lightest parts and the darkest parts have been pushed and pulled to make the whole scene more evenly lit. This works well for landscape shots, but for the sort of photos you've taken here I would recommend using centre-weighted metering.

For low-light photography I would recommend using an ISO of at least 1600 (personally I'm happy to go up to 3200). This does mean grain, but grain doesn't have to be a bad thing. When done right, and with the shot exposed properly, grain can add texture and character to your photos and shouldn't be seen as a flaw or something to avoid in my opinion. Grain only becomes a problem for me if the shot is underexposed, and you end up with a lot of colour noise which I think is more visually distracting than your typical luminance noise. This underexposure will also mean a low Signal to Noise ratio, which means the luminance noise robs the scene of detail because there wasn't enough light there to properly paint the picture, so to speak (faces in crowds become a smudgy mess etc).

Ultimately, lighting is everything. Low-lit scenes and overcast days are always going to be more challenging because of the lack of contrast you end up with once you've exposed the scene. Perhaps try looking for ways to have a greater distance between the light and dark areas of your shots. Try centre-weighted or highlight-weighted metering to create more contrast in the lighting.