2 - Check for motivators: See if the dog is motivated by treats, balls, squeaky toys, or a rag or bite tug. A more motivated dog is easier to train.
3 - Check for fear: See if the dog is afraid of certain actions or noises. If you go to pet the dog over the head and it drops its head or pins the ears back and squints, it may have been beaten before (not a reason to not adopt) and may need some additional positive reinforcement to not be afraid of hands petting the dog.
Abrupt sounds will help test a dog of fear of certain noises. For example, clapping 2 wooden pieces together making a sharp sound without the dog seeing you do so will help you test the dog for fears of certain sounds. Dogs that are afraid of those sounds may often have problems with thunder or confidence issues around abrupt sounds (again, not a reason to not adopt as all this may be reconditioned in the dog with proper behavior modification).
4 - Check for the dog's reception to other people and dogs: First see how the dog reacts to people approaching its cage and then see how the dog reacts to other people approaching the dog while on a leash. The best reaction is one where the dog doesn't even care if there are other people or dogs in front of them. Shy or reserved dogs may need some additional socialization work. Extremely hyper-friendly dogs may need some impulse control work.
5 - When possible, have the dog professionally evaluated by your veterinarian and / or local behaviorist for more detailed assessment of the dog's psychological make-up and training potential.