Tahoe Donner has extensive hiking trails around the perimeter (walking distance from the house), up to Donner Ridge (from Glacier Way parking lot), down into the Euer Valley (from the end of Alder Creek Rd), and a very nice loop along Trout Creek that starts across the street from the Northwoods Rec Center. See the trail page here.
Commemorative Emigrant Trail runs along Alder Creek (accessed from Schussing Way and Alder Creek Road, just north of Tahoe Donner). Cross Alder Creek Road and follow the trail toward the Donner Picnic Area and Prosser Reservoir east of Hwy 89. Part way along, turn left and go up some banked S-turns built for mountain biking for nice views to the NE. “Dog Beach” is on Donner Lake, accessed through Donner Memorial State Park, next to the Tahoe Donner Beach Marina (tho no dogs are allowed in the marina), on Donner Pass Road.A biking trail along the Truckee River is found from the very end of East River Street (cross the pedestrian bridge) or from the Truckee Regional Park on Brockway (at the E end of town, turn right on Bridge Street, cross the river where it becomes Brockway, and the park is on the left.) Work your way down to the lowest elevation parking lot, where a paved trail begins toward the east, the Legacy Bike Trail.
Truckee River Legacy Trail website
From the Legacy Trail, shortly after a pedestrian bridge back to town, watch for a foot-path down to the river on your left. This path parallels the river for several miles, and is a safe trail for dogs, far from bikes and cars, and the river moves relatively gently (in the late summer at least).
A large tract around Martis Creek is great for dogs, hiking, and biking. From Downtown: I-80 east past downtown Truckee, take the exit for SR 267, turn right, S, toward Northstar. After the airport, watch for “Wildlife viewing area” sign and turn right into the parking area.
A wonderful dog-friendly trail goes from the Alpine Meadows Road up to the Pacific Crest Trail and several small lakes, between Alpine Meadows and Squaw Valley ski areas.
Regional hiking and biking guide books and maps are on the bookshelf in the living room.
One other resource is this list of hikes under 5 miles: