Is stage 2 breast cancer treatable with radiation and chemo, or will I need surgery?

You will need all. Breast cancer in stage 2 still needs a lumpectomy and nodal evaluation which is surgery. Afterward chemo and radiation is often recommended.
Surgery is needed. Currently, surgery is an essential and starting point for the treatment of early (stage 1 or 2) breast cancer. Surgery removes bulk tumor and involved lymph nodes and also helps in accurate staging leading to the correct use of follow-up chemotherapy, radiotherapy, or hormone therapy. Less surgery, usually involving sentinel node surgery for the armpit nodes is currently practiced.
Surgery is needed. In just about all stages of curablle breast cancer, but the kind has changed. While you can avoid mastectomy most of the time, some women opt for bilateral mastectomy to avoid screening (no evidence of improved mortality) lumpectomy is as good if not better than mastectomy with properly selected cases. Arm pit surgery for nodes has been questioned, but sentinel node is standard.