Pre-heat grill to high heat. Toss asparagus with olive oil, salt and pepper to taste. Grill until tender, about 5-6 minutes, rotating once. Remove asparagus from grill and place on cutting board. Cut asparagus into bite-sized pieces.
On a large platter, scatter the arugula and top with red onion and grilled asparagus. Drizzle with olive oil, lemon juice, salt and pepper. Crumble goat cheese on top before serving.