you can walk across the city in 15 min. A lot of buses run along Errazuriz for 300-450 pesos.
Gonforterio: small pale-ontological site with bones from a mammoth like creature. Two trailers make up the visitor center with artifacts and displays about ice age animals. Guides are really helpful and will give you a tour. Free; in Spanish. At the end of calle rio elqui, across the river from Parque Chuyaca (which has ice age animal statues)
Several historical houses can be found Ave. Juan Mackenna near Calle Cochrane
San Juan de la Costa: Several awesome beaches are about 1.5 hours from Osorno. Off the beaten track, beautiful views, quaint villages. Really a traveling goldmine. You can get a colectivo for 1,500 pesos from the terminal on the far side of town (calles tarapaca and temuco). First one is at 7:30.
Parque Puyehuy: hot springs, hikes, waterfalls. Free hotsprings can be found along the riverbank of aguas calientes. From where the micro drops you off, go through the parking lot, towards the outdoor pool that's part of the paid complex. You'll cross a pedestrian bridge, take a right after the bridge and in about 100 yards you'll find the pools after the picnic benches. Catch a colectivo to aguas calientes for about 2,200 pesos.
Entre Lagos: A 1,500 peso ride takes you to the small town of entre lagos. Very tranquil, good views of volcanos. There's a park near the water front that has crazy/dangerous slides, good trees to climb, and small crayfish type animals you can catch under the rocks.
Pick blackberries: wild blackberries grow all over the city, big patches can be found in Parque Chyuaga.
Osorno is centrally located to many great destinations including Bahia Mansa (and other great coast towns), Parque Puyehuy, Frutillar and Puerto Varas, and Valdivia. See get out for more info
urban exploring: Some abandoned buildings can be found near the train tracks and on ave Pilauco near Parque Cuarto Centenario.
Pueblito Artisanal offers hand woven hats and gloves for fair prices.
Several grocery stores are located around the bus station on Pratt and Colón
Colectivos to most destinations can be found the the terminal on the intersection of Errazuriz and Angulo. Buses to Parque Puyehue are one block up on Errazuriz and Colón. All these buses can also be caught on the way out of town at the bus stops along errazuriz and Recabarren. First departures are usually around 6:30 am.
Colectivos to the beach are found at the terminal next to the Rahue market. The first one is at 7:30 am and they leave somewhat erraticly in the winter, usually around every 45 min. A schedule is posted by the wall dividing some market space from the large bus parking area.
Cruz Del Sur has direct service to Chiloé for 6,000 pesos (april 2014)