What are the best ways to manage anxiety and panic without meds?

Many. I agree with dr. Fowler's suggestions, and want to add the option of psychotherapy with a skilled professional. Cognitive behavioral therapy is one -- but not the only -- form of therapy that can help anxiety and panic. Choosing the best methods to help, depends on the clinical situation. Some people do respond well to meds that manage panic attacks, but therapy is important in getting well.
Alternative tx. You might consider options like acupuncture, yoga, deep breathing exercises or progressive muscle relaxation and ensure that you are not using alcohol, tobacco or caffiene to excess. Take care.