2 types of penguins can be seen in two moderately priced day trips from Punta Arenas, Chile. Punta Arenas is an easy cheap flight from Santiago and there are accommodations to suit all styles and budgets.There are a few other sites in the area and it is a good starting point to then take the bus to Puerto Natales to explore Torres Del Paine. More about planning a trip in this area here.
The most popular penguin trip from Punta Arenas is to Isla Magdalena to see the Magellanic penguins there are said to be over 100,000 penguins on this island. These penguins are similar to the African penguin in look and sound. To get there it is a 2 hour boat from Punta Arenas then you have an hour or so on the island and the boat back. Price is around 62.000 Pesos ($75USD) for boat and entry plus English and Spanish narration. I have not done this tour since November 2003.
There is a king penguin colony on Tierra Del Fuego called Pingüino Rey Park . I only learned about this last year. The penguins are there year round it is a smaller colony and you cannot get very close but is the easiest place I know of to see King Penguins. Otherwise they are on Antarctic islands.
The tour is all day, given how many times they warned us about the bumpy roads I gather many people complain. Some additional stops are made but the penguins are the highlight and it is a lot of travel. I went with a company called Denomades which worked out well. Cost was 50.000 ($60) Chilean Pesos not including the park fee or lunch. It did include transportation and pick up at lodging for the 12 hour day trip. You take a car ferry to and from the island Tierra del Fuego. I looked into doing this independently but the timing and coordinating with the ferries made it difficult. I did this tour in January 2020.
Tip: Bring binoculars and a camera with a lens if you have one. The King penguins can be far from the boardwalk.
This is the best Map I have ever seen explaining the penguins in Patagonia. I have not visited most of these but hope to one day!
Punta Arenas is also a starting point for longer local cruises and even flights to Antarctica. You will see even more penguins with these options but they will be much more expensive.