Pulling
Yu can ask whether the shelter can get you in touch with local volunteers. That is a good starting point. You can find area rescues on Petfinder and contact them too. You can also request help on this or other boards - and you will have to be prepared to answer questions.
Transport
Paid transport is the easiest unless you are an established rescue that already has transport contacts. Also people are more likely to volunteer for established rescues or transport coordinators they worked with in the past. You can try to ask a transport coordinator to help out, you will probably have to answer questions. Personal connections and friends would be one option. Flying a dog is not unreasonably expensive either - usually.
Any transport will require vaccinations to be up-to-date and the dog to be flea and parasite-free and reasonably clean. Transport accross state borders requires health certificate (different for air and ground transports, you need to check with the transporter).
This is all in a nutshell.