Tanjung Pinang's location in the Riau Archipelago puts several interesting destinations within easy ferry range.
Penyengat Island is the closest and most rewarding trip -- just a few minutes by sampan from the harbour. This tiny island (2.5 square kilometres) was once the seat of the Riau-Lingga Sultanate and is packed with history: the Sultan of Riau Grand Mosque, built with egg whites and lime, royal tombs, the grave of grammarian Raja Ali Haji, and the ruins of old palaces. The island has about 300 residents and can be explored on foot in a few hours.
Batam is a short ferry hop away and offers a contrast -- it is more developed, with water sports, go-kart racing, and a busy nightlife scene around the Waterfront City area.
Singapore is two hours by ferry and makes for a full but feasible day trip. Book your return ticket in advance, especially on weekends and holidays when ferries fill up fast.