Lansing's "Endurance" is the classic starting point. If you read that, Lennard Bickel's "Shackleton's Forgotten Men" is a must-read too because it tells the rest of the story. Highly recommended.

For a different approach to the Shackleton legend, try Caroline Alexander's "The Endurance". It has lots of wonderful photography from Frank Hurley, the expedition photographer.

Once you think Shackleton was the toughest, most rugged explorer, read "Mawson's Will", also by Bickel. It will give you nightmares.

To see how one man can fumble an expedition despite best intents, try Adm. Byrd's "Alone".

The ultimate classic, of course, is Apsley Cherry-Gerrard's "The Worst Journey in the World." This is a tome, kind of Moby Dick-like in its detailed analysis of the environment; but it's also a diary of real-life events. If you can slog through the trivia and tedious bits, there's some gem passages in there.

I suggest Alfred Lansing's telling of sir Shackleton's expedition to the South pole (called "Endurance").

It's an interesting area to read about. People bring a lot of biases to the table on this subject. It is basically a split between people who lionize the British explorers - Scott and Shackleton (primarily), and people who laud the Norwegian - Amundsen and Nansen (primarily).
I would also suggest going beyond just the Antarctic expeditions. You will learn a lot from Nansen's journeys and Amundsen's early journeys as well.
"The Last Place on Earth" by Huntford is a wonderful book, but has a strong bias to Amundsen.

I had someone highly recommend a book to me just yesterday on this--except in the other direction, about the Arctic: In the Kingdom of Ice: The Grand and Terrible Polar Voyage of the U.S.S. Jeanette. Basically the story is (forgive errors, I haven't yet read the book myself, but I plan to, so I can only repeat what was described to me) that there was a theory that warm ocean currents made it possible to sail with a steamship all the way to the North pole. The New York Herald commissioned the Jeanette expedition to do just that in 1879. They got pretty far, got stuck in the ice, and stayed for 2 years until the ice finally crushed their ship. The crew then took 3 small boats and walked something like a thousand miles across the ice floe (they were walking south, while the ice was drifting north, and ended up farther north than when they started). Then they sailed to Siberia. A storm hit on the way, one boat presumably sunk, and the other 2 made it to the Lena Delta in Siberia, but were separated from each other in the storm. One group made it to civilization and safety, but the other group (that included the captain of the expedition) essentially starved/froze to death in the middle of nowhere Siberia. The group that made it then went searching for the other group and found them, but too late.

It sounds like a tragic, page-turner of a story. And from what I understand, the trip was surprisingly well documented too. The scientific records kept on the expedition largely survived (astounding that they managed to protect their logbooks and such through all of that), at least 2 of the survivors later wrote books about the experience (there were only like a dozen who survived), and since the expedition had been commissioned by a newspaper, they had a reporter on board who was writing about the expedition and sending out dispatches when he was able to.