The first paragraph of the story paints Phoenix as a small, old woman. She "was very old and small" and used a cane made from an umbrella with which she tapped the ground in front of her making a noise "like the chirping of a solitary little bird." This connection of Phoenix with a bird, in this sentence, makes her seem small and ineffectual. However, as the author develops the character, the reader will learn that Phoenix is quite resilient and resourceful. Thus, the name Phoenix rings true as it refers to the mythical bird which would rise from its ashes.
In the third paragraph, Welty begins to show Phoenix's abilities, noting that her handling of her cane was as "limber as a buggy whip." Although she mistakes objects (thinks a thorny bush is a green bush, supposes the scarecrow is a ghost), she perseveres. Phoenix is not scared when the hunter rudely points the gun in her face. She even manages to snatch a nickel the hunter had dropped; a bit of poetic justice considering the man was condescending and lied about having no money.
Phoenix also shows her generosity. She doesn't make the journey for her own benefit: her resilience and the purpose for her journey is to care for her grandson.
She doesn't acknowledge that the medicine she gets is for "charity" but she's too wise not to accept help when it is offered. She takes a nickel from the attendant and speaks quite deliberately and precisely about what she is going to do from there:
"I going to the store and buy my child a little windmill they sells, made out of paper. He going to find it hard to believe there such a thing in the world. I'll march myself back where he waiting, holding it straight up in this hand."
At the beginning of the story, Phoenix is portrayed as a weak, old woman. But as the story unfolds, the reader learns how resilient, resourceful, and determined she is.