Most first-time visitors attempt the W Trek: a five-day, four-refugio traverse of the park's three main valleys. It is not technical, but it is unremittingly exposed, and the wind can knock a grown adult over on a bad day. Train for it like you would for a marathon and book refugios six months ahead.
Weather is the experience
In Patagonia, weather is not the thing to be managed on the way to the scenery — it is the scenery. The clouds move faster than they have any right to; the light turns every fifteen minutes; you will be rained on, snowed on, and blown dry within a single afternoon. Bring the right shells and the right attitude and it's thrilling. Bring neither and it's miserable.
If you're not hiking
You don't have to walk the W to experience the park. A three-day stay at an estancia on the southern edge gives you day hikes to Laguna Amarga, boat access to Grey Glacier, and enough open sky to understand the scale without the suffering. The food at the estancias is often better than at the refugios, too.
Getting there
Fly into Puerto Natales via Santiago and Punta Arenas. The three-hour shuttle into the park is part of the experience — you'll see the landscape change from steppe to glacier to forest in a single morning. Do not drive the last stretch yourself; the wind is legitimately dangerous for small cars.