First, welcome.
Second, there's tons of info out there on Los Glaciares. Books, even.
You'll be there in the high (crowded) season, so be prepared for that whether camping or staying in a hotel.
A walk on Perito Moreno glacier is not to be missed. There are short (2-3 hours) and longer hikes (6 hours?). You can hook up with a guide at the shack a day or two before to make a reservation, but you may also be able to just walk up and get with a group. Crampons are supplied. It's a short boat ride from a nearby hotel (can't remember the name).
In the northern part of the park there's Lago Viedma, El Chalten (get supplies there), Fitz Roy, Cerro Torre.
In the southern portion there's Lago Argentina, Perito Moreno glacier, El Calafate (get supplies there) and more.
From El Chalten a hike to Fitz Roy is worth it. For my money, though, I consider nearby Cerro Torre to be the most beautiful/strangest group (look up "pluton"). It's a nice day hike (12 miles?) from town to the mountain and back, but there are multiple trails if you want to add more days.
Buses are the best way to get around over long distances. They may not run every day, even during the high season. The roads will almost all be dirt and rocks, sometimes narrow, and you can bet on 2-6 hour rides (they don't go fast due to the road conditions).
It's an awesomely beautiful part of the world.
Link 1Link 2Map (I just found it on amazon - don't know how accurate/useful it is)
If you don't already know some usable spanish, I highly suggest signing up for a couple of courses - two semesters will be enough. Oftentimes a local community college will have non-credit but highly practical courses for very little money, usually at night. These courses tend to focus on essential vocabulary and reading/writing/speaking, rather than the more formal grammar of the credit courses (you'll be asking for directions, not translating for the State Department).
Be prepared for wind! I haven't been to that area in about 5 years, but when I did I experienced at least two days of sustained 40mph winds with 60mph gusts. I met a woman who said she fell down when the wind SLOWED DOWN for a few seconds. She was leaning over so far that when the wind suddenly abated she just fell over.
ETA: I was farther north last year, and it was about 10F warmer than normal at night and 10-20F warmer than normal during the day. I was told that it's been that way for the last few years. Don't know if the same is true where you're headed, but it'd be worth trying to get some local info on - actual recorded temps, not historical records or predictions for the next week.
From what I know El Nino/La Nina aren't huge factors there, but El Nino years can be a bit warmer and drier. I saw virtually no rain in 2+ weeks, but this last year I was farther north in the Aysen region, closer to Cochrane and Coyhaique, where as you will be further south in the Magallenes region.
Perito Moreno glacier from the trail
Perito Moreno glacier detail
