Tips for photography in dull grey weather?

[fallbayuk]

I've only recently got a DSLR but the weather's been crap ever since it arrived as is normal in this country for about 8 months of the year. In such dull grey conditions, what does everyone do to try and create the best exposure possible? Is it a case of just bumping up the ISO? And generally speaking what do you do in lightroom to improve your photos taken in this sort of weather?

I'm off to Paris next week and the weathers pretty similar over there at the moment so want to not screw up as many photos as possible.

 

[fallbayuk]

Cheat. Photoshop a sky in after altering the WB

I'm mostly worried about minimising camera shake as I seem to be a bit of a shakey person. If it's dull, my shakiness is more apparent because of the slower shutter speeds. That's what I think is happening anyway, hence I thought upping the ISO would help.