You should be fine Nellie. Just be sensible - choose places where there are other people as well. Most dog walkers are friendly, and their dogs are generally fine too. Don't go for dark alleys, tow paths where there is nobody else, etc. Take your mobile phone with you. Avoid pavements if possible - they're a bit hard on your knees. Tarmac, local parks and trails offer a softer running surface.
And especially, join your local running club - that way you don't run alone. There's a link at the top of this page to find running clubs in your local area. And your gym may well have an outdoors running group already.
Good luck, and enjoy!