I run with my dogs, too, and as Kima has said, it forces you out. Here on the North-east Scottish coast we get some horrendous weather - in fact, there's a gale blowing right now, in mid-June! - and for large chunks of the winter it's not light till gone 9am and dark again by 3pm. How much do you think I enjoy taking out two large beasts in a Force 8 on a cold, wet, November day?!
But my reasoning is that they have to be exercised, whatever the weather, and you can go a lot further a lot more quickly by running than walking! That gets me out the door!
Seriously though, running is a wonderful stress-buster along with everything else. As others have said, set achievable targets to start with and work up. When I first started 20 years ago -after quitting smoking 30 a day - I did the run/walk thing and took it from there.
I also joined a running club, which is another great way of forcing yourself out on a foul night - that was in the north of England, which also does a fine line in wet, horrid weather! There are loads of other benefits to be had from being in a club, of course, particularly learning from the experience of others.
Another good way of keeping focused is to pick a race - a 5k is a good one to start with - and work towards it. The sense of achievement the first time you cross the finish line is really brilliant!
Nothing much else to add really, except to echo the stretching message and the shoes message. You can't get away with skimping on either.
Good luck and enjoy!