Punta Arenas is a port city in the Southern tip of Chile. They consider being the Southest city in the world, but technically Ushuaia in Argentina is more South then Punta Arenas.
The city is always cold even in the summer time (opposite to the Northern hemisphere; so when it is the Winter in North, it is summer in the South, but you know that...). They have unusual strong winds and even have some ropes on some of the boardwalk in the city center to grasp a grip on the windiest days! It rains a lot too; the sunny days are rare in this area of the world. That tends to give a depressing atmosphere to the place, although if you are lucky and have a sunny day, the sun shines strong and UV rays can give you easily a sunburn since the ozone layer in this area of the world is thin. That gives a rare blue colored sky and some setting unique in the world. Wear layers as it may cold in the morning and then warm up quickly in the middle of the day when the sun comes out. Your pictures will have this special look that shows you had been in a far area of the world.
Punta Arenas really gives you the feeling you reached the end of the world. If you venture a bit out of the city, you will get that unusual feeling of being so far from civilization and a feeling of loneliness...
The first attempt of settlements in the area were failures and the people starved to death, even though they had the nicest King Crabs in the surrounding waters; the people were scared of their look and thought they were poisonous!
There is a tax free zone in the city if you feel like trying your luck for shopping, but honestly I haven't found the items cheaper than elsewhere.
The port is within walking distance to the city center and it is easy to visit the city by foot. Plaza de Armas is the square in the middle of the city and you have some handicraft sellers in that square. You can buy products from the region as the typical alpaca wool garments.
Maggiorino Borgatello Salesian Museum is one of the museum to visit with some native history. It is quite small and if you are not a museum fan, I wouldn't recommend it.
Trips to do:
By far the best day trip to do is sailing to the Magdalena Island with its huge colony of Magellanic penguins. You can do it by mainly by ferry or some little boats that some agency can organize (if you can do this, it is faster and not much more expensive!). It takes close to 6 hours depending on the sea condition to do this tour and usually out of the 6 hours, you spend 1 hour on the island walking on some roped trail to the lighthouse and back to where the ferry land.
The condition of the ferries depend. They may use some very old ferry that they charter out of service for the tours or there are some new ones in use. The most important factor to consider is the sea conditions: if it is rough seas, expect some roller coaster ride back and forth for hours... Sea sickness to be taken 1 hour ahead for the ones prone to sea sickness!
There is another shorter penguin trip to Otway Sound, which is made by bus, although I do not recommend this trip at all! You could see between 10 to 50 penguins in the distance hidden in the grass. There is a long walk where the cold wind blow strong, so be ready! I use to get more complaints and disappointment from this tour than any other in this region.
Fort Bulnes is a reconstruction of the original settler's fortress and to my opinion is not much worth the long ride unless you are a fan of forts. Port Famine is usually coupled with this visit; it is the original settler's site, although there is only a rocky beach left with no interesting sights except the sea to admire. This was never a big hit as an excursion.
In the same spirit, the cemetery is another "site", although I wouldn't recommend it unless you are a fan of cemeteries or you have distant relatives buried there.
The estancias in the surrounding areas are interesting, especially the far ones like Fitz Roy. It is a long ride to get there, but the feeling you get while there is really somewhat exotic with the remoteness of the area... imagine sitting in the living room with a fire place cracking, the windows opening to an endless scenery and the cold wind whistling trying to penetrate this last cozy little place at the end of the world... They also have some llamas and some sheep shearing demonstration which can be entertaining. Of course, the lamb of Patagonia is always at its best on their fine BBQ!
They do have a lot German immigrants in Chile and some fine brewery. There is the Austral beer brewery that you can visit or at least while you are visiting. Seafood is really the culinary delight of the area. Find recommendations for the local seafood restaurants such as "La Marmita" or "Brocolino".
For the one braving the strong winds, Mount Fenton is reachable by a short taxi ride and you can take the cable car up and come down by zip-line. You will get a panoramic view of the city from this hill.
Of course, you can fly to Torres del Paine in one day and get a windy ride or you can travel there for a few days excursion. There are also some flights to Antarctica although they get cancelled often for weather reasons and they only can fly for a short period in the summer.