The best advice I can give you is ... be honest.
Tell the agent that you're moving on your own for the first time. Offer a pet bond and give a pet resume. If you can, offer a couple of months rent upfront and even consider a guarantor (like a parent, or your boss).
Have your applications complete. Don't miss any sections. Provide good references. Because you can't provide rental ones, provide one from your boss, neighbours showing you're a quiet person and that your dog isn't a nuisance etc. Even a reference from your doctor saying how long he's known you can help.
Just be honest, respectful, dress nicely and use your manners. First impressions count!
Good luck 🙂