I'd definately not rush into renting as more often than not leases are for a year here and so if it ends up you hate the area or the house/landlord turns out to be awful you can be stuck there till you find someone else to take on the lease or it ends.
I'd not go via Gumtree but that is me. Plenty of dodgy people out there for new arrivals to get ripped off. Sure you get some decent ones but honestly, going via an agent for your first rental is probably not a bad idea. At least you know what is what and the process is all there for you to walk through.
Also look out on here for people breaking their leases (usually as moving into own house or moving areas) as they will hopefully vouch for the house and landlord.
And don't be put off about pets. If a property states 'No Pets' then I'd cross it off and not bother looking. If it doesn't say anything either way, its worth a punt but be prepared to pay extra bond to help seal the deal. If its Pets Allowed then the house may well show signs of previous pets or the garden may not be great if dog has lived outside. But you'd be fine for keeping your dog there.