Algiers has a small metro system — just one line running from the southern suburbs to the centre — and a network of ETUSA buses that covers most of the city until around midnight. Both are cheap. Taxis are plentiful and inexpensive by European standards, though you should agree a fare before getting in.
The city is walkable if you are fit: the centre climbs steeply from the waterfront up towards the Casbah and the residential hills. Walking is the best way to explore the old town, but for longer distances or after dark a taxi is the better choice.