To and from the airport, Transfer Aeropuerto Valdivia provides an on-demand minibus service, but the most economical way is catching any Buses Pirihueico bus to Panguipulli from the bus terminal, which will drop you at the airport (CH$1000, 6:30am to 8:35pm). A taxi from the bus terminal costs CH$20,000 (an Uber runs CH$12,500 to CH$17,500 from the center). The trip takes about 30 minutes.
From the bus terminal to Plaza de la República, head to the south end of the terminal and up the escalators to the 3rd-floor exit onto Carlos Anwandter, where you can take any green or yellow colectivo (CH$500 by day, CH$600 at night) or bus 3 (CH$450), which will drop you on Chacabuco, one block north of the plaza.
For Niebla, take colectivos or Micro (minibus) 20. From the Muelle Niebla, small boats go back and forth to Corral every 20 minutes or so from 8am to 9pm (CH$800) and ply the waters to Isla Mancera every 20 minutes a few times daily (CH$300; 9am, 12:30pm, 2:30pm and 5:30pm Monday to Saturday, 10:30am, noon and 4pm Sunday). There is also a larger ferry that can carry cars to Corral (every hour, 8am to midnight).
Hertz, among other agencies, rents cars in the city.