What sort of activities should I avoid if I have an old ankle ligament injury?

Avoid uneven surface. Most ankle sprains occur when the foot rolls inward affecting the outside ligaments. Ice, elevate and compression with acute injuries. Supportive brace until healing. Restoration of range of motion, strength and proprioception (balance) is key to prevention of injury. I would emphasize one leg balance and increase time until you can do it for one minute. Advanced activity stand on foam pillow.
None. If it was treated and healed properly you should not have to avoid any exercises. You may want to consider using an ankle brace for protection.

Related Questions

How do doctors treat ankle ligament injury?

Depends on severity. It depends on if the ligaments are strained, partially or fully torn, and how many ligaments are injured. Treatment may include rest, ice, elastic ankle wraps, antiinflammatories and rest. More severe injuries may require immobilization or bracing. Physical therapy or surgery might be needed. See a podiatrist or orthopedist for more specific treatment recommendations.