Halifax is a military town. It was founded because the English needed a naval base to offset the french base at Louisbourg. So there's a lot of military history to be seen in the city, mainly old outposts that have never been tested. Go see York Redoubt; the magnificent view of the open harbour is amazing and understand why Halifax would have been a formidable target. Take a hike along Duncan's Cove. It's not an official maintained hike, but very popular. It starts at Ketch Harbour and winds down to Duncan's cove. Excellent way to spend 2-3 hours and you get to see WWII gun placements.
Many people miss York Redoubt which is located just outside of Halifax. It is another historical site located at the mouth of the Harbour. It provides a great view of the peninsula and more of Halifax's great history. There are also many good walking trails around the fort.