There is no right way to become an ornithologist — actually, your degree wouldn’t be in ornithology per se, but instead wildlife, environmental biology, natural resources, or something similar. Ornithology is just the focus that you would aim for under these broad umbrellas. Look and apply for job or internship opportunities related to working with avian species (and make sure they are PAID)! If you want to go to grad school, building up your portfolio with experience will help in the long run. Most importantly, your love for wildlife, and nature, will see you through on your journey.
Unless I have worked closely with you over a few months or years (i.e. mentored you on a project, continuously helping with my field work) I ask that you please do not list me as a reference on your application. I want (and need) to make sure I know you well enough and can speak to your work ethic. I want my recommendation to do you justice!