The city itself has 120,000 people. It has a cute square plaza where people hang out in the day and couples at night. It has some great old homes with the restored and original decor. However other than this not much else to do. Buying fresh fish is a treat though as so so cheap.
Tourist information is a challenge. It is tough to find an english speaker, but even tougher to find someone who even knows what is going on in the city. Well at least that is our experience of the 10 odd tourist places we went to. Having said that there are many people more than willing to try to help you out.
To get out of Punta Arenas, the easiest way is by flight. Air Comet, Sky Airlines and Lan Chile are the carriers. The airport is 20 minutes away by taxi and it costs around 5000 pesos for the whole car. You can take a minu bus called ´Transfer for 2000 pesos each´. So depends on your budget.
You can get to Punta Arenas by plane, or by bus say from Puerto Natales. 3 hours bus ride for 4000 to 5000 pesos depending on the bus company. There are a few companies not all in the same area so you have to go searching around the city for it. However it is so small you can find everything in about 30 minutes.
You can get by bus(with ferry inlcuded) to Ushuaia. This is the southern most city in the world situated in Argentina. You catch the bus in the mornings for 24,000 pesos one way. Ride takes 10 hours.
There are many hostels in the city. You can find one for 4000 pesos a night each with free internet and breakfast (negotiating is possible on prices). We walked the streets looking for one without a pre-booking. No probs.