The biggest day-trip pull from Shangri-La is Tiger Leaping Gorge, two hours south on the road to Lijiang — one of the deepest river gorges in the world, with an excellent two-day high trail along its northern rim. Going on foot is the way to do it: from the trailhead at Qiaotou you walk over two days to Tina's Guesthouse at the gorge bottom, with simple guesthouses along the route.
For shorter trips, Pudacuo National Park and Napa Lake are both within an hour, Shika Snow Mountain is at the town's back door, and the Songzanlin Monastery is twenty minutes from the Old Town and gets nearly every visitor.
Pushing on, the road north to Deqin and the Meili Snow Mountain (about eight hours) is the gateway to the eastern Tibetan plateau. Most travellers continue this way rather than treating Deqin as a day trip — there is plenty to do up there.