Shigatse is small enough to walk between the monastery, the old quarter, and the dzong in an hour. The city has expanded with Chinese investment and the Chinese commercial district is mostly characterless, but the sights are clustered in a compact area around the monastery.
Bicycle hire is available near the main hotel area and makes sense for exploring the outskirts — Narthang Monastery to the west and Shalu Monastery to the south are both reachable by bike on reasonable roads, though Shalu involves some rougher track. Most guesthouses and tour agencies can arrange vehicles for day trips further afield.
Altitude is a factor at 3,840 metres. Arriving from Lhasa (3,650 m), most people acclimatise reasonably well; arriving from lower elevations can mean headaches and fatigue on the first day. Drink water, don't over-exert, and allow a day or two before attempting any significant hikes.