This study will evaluate the safety and effectiveness of a staged approach to therapy for Wegener's granulomatosis and other systemic vasculitides using prednisone plus cyclophosphamide followed by methotrexate. Vasculitides involve inflammation of blood vessels (vasculitis) that may affect the brain, nerves, eyes, sinuses, lungs, kidneys, intestinal tract, skin, joints, heart and other sites. Standard treatment with prednisone and cyclophosphamide is very effective, but has significant toxicity (adverse side effects). Methotrexate is also an effective treatment and is less toxic, but it is associated with a higher rate of disease recurrence and has not been used in patients with severe lung or kidney disease. Staged therapy using cyclophosphamide first and then methotrexate may provide better results for overall safety and effectiveness.