Three examples of symbolism

Candy's dog represents Candy. It is old, arthritic, and has little practical use. Candy muses about whether they will shoot him when he is no good. There is a little foreshadowing here as well with a similar fate for Lennie.

Lennie's puppy symbolizes himself. It personifies a friend that Lennie desperately needs yet kills it. Similarly George must kill his fiend Lennie as well. The puppy also represents innocent emotional/intillectual frailty as Lennie has as well.