Hitchhiking - Haciendo dedo
Chile is a great country to hitchhike in. You do not need to use a sign with the name of your intended destination. Showing an extended thumb is enough to entice a driver to pull over.
It is common to wait 1 to 2 hours for a car to stop for you on the Carretera Austral. Be aware that not many people drive down the Carretera Austral at night, as most of the highway is unpaved, and it does not have reflectors or lights.
At times, there can be a lot of competition amongst hitchhikers to get out of some of the towns in Chilean Patagonia due to a low volume of traffic. Some of the toughest towns to hitchhike out of being Chile Chico and La Junta.
A ferry across the Strait of Magellan to Tierra del Fuego
There are a number of private bus companies that operate throughout the country, the most popular operations are Pullman, Andes Mar, and Tur Bus. Some Chileans may warn you about using Tur Bus for very long distance trips.
Note that long distance buses may not operate daily on certain routes. It is possible that there may only be a handful of buses between some cities each week.
There are some bus operators that have services to smaller towns and villages. You will need to visit the information center or bus station in whatever town you are in to find out what options are available.
Trains are not a great option when traveling south from Santiago, buses are a better choice.
There are a number of people who travel by bicycle from Santiago all the way down to Punta Arenas or further south to Tierra del Fuego. It may be possible to include fly fishing on a cycling journey.
Buying/Renting a Vehicle
There are guides on the internet that explain how you can buy a vehicle in Chile as a foreigner. One can be found here.
With a vehicle of your own, you can travel about much easier, allowing for more fishing time. But having a vehicle will make your trip more expensive and could prove to be a headache if you have any breakdowns. Be aware that many of Chile's southern roads are unpaved and in rough shape. If you purchase a vehicle in Chile as a foreigner, you will have an easier time transporting your vehicle to other South American countries, compared to if you were to purchase a vehicle in Argentina.
It can take a considerable amount of time to travel overland from Santiago to Coyhaique or even further south to Punta Arenas. It may make sense, from time and cost perspectives, to fly to one of these cities from Santiago and then spend your time fishing and traveling north towards the Capital.