Aerolineas Argentinas has daily flights to/from Bariloche,Buenos Aires and Ushuaia. LAN also flies on Monday, Wednesday, and Friday from Buenos Aires. On-line booking and e-tickets are available. During peak season, flights are usually fully booked with tourist groups. Make sure you book your flight well in advance.
There are also charter flights between Puerto Natales and El Calafate operated by DAP. These flights from the tiny airport of Puerto Natales can be very scenic and let you cross the Andes in about 90 minutes and no road travel hassle (which however in right doses is something not to be missed in Patagonia).
The only other possibility is to arrive by bus. But this means that you will spend quite some time in the bus on rough roads. Due to the dirty streets, especially when it rains, views from buses vanish fast...
A quite remarkable trip is the bus connection between El Calafate and Bariloche along Ruta 40. It is mainly operated for the benefit of tourists by the Agency Chaltentravel (directly only the Calafate-Perito Moreno stage), since people of Calafate and Bariloche have very small ties between each other, which does not sound strange after you have made the journey and seen the state of the road.
Views, if the weather is good, are in places stunning, especially between Calafate and Perito Moreno city, less between Perito Moreno and Bariloche. You must however be prepared for a long (10+ hours for 2 1/2 days) and rough trip in a small van. If you are not the kind of person who enjoys slow changes in the landscape and the beauty of lonely places, do not attempt this trip or you will regret it. Of course, if you really like this sort of thing, you will never forget the trip.
The bus trip will let you pass (and most likely stay and enjoy) through El Chalten and give an opportunity to stay in Perito Moreno from where you can organise a (long) ride to visit the "Cueva de las manos pintadas".
Make sure you book your bus ticket out of this town, when you get there, or you can get stuck there for a couple of days.
A must, the Perito Moreno Glacier, 80km (50miles) away from El Calafate. You can see the enormous glacier just meters away from you, and listen all the sounds that the cracking ice makes 24 hours a day. And when a front wall of the glacier falls down, well, it is just amazing! It is the main point of tourism. Viewing platforms and easy trails (with quite a number of steps, though) make the visit and the observation easy and pleasant. You can visit many ranches (estancias), where you can enjoy the wild nature, perfect meals, and stay there in the middle of nowhere, relaxing from the big cities rush. Or take a boat excursion and see many of the glaciers and places in the National Park Los Glaciares. The colors and shapes of the ice-bergs floating in the lake are really awesome and inspiring, making a boat ride in a sunny day a fantastic day out. Another perfect point, is Onelli Bay, one of the most romantics sites here. Just bring your lunch box, a bottle of champagne, and chill it in the lake, or just add a piece of pure crystal ice. Wild nature can be seen just walking a few blocks away from main street, like in El Cañadón, or at Laguna Nimez, where you can see all kind of patagonic birds feet away from you. Wildlife and views, especially at dusk, make Laguna Nimez a must see, perfect if you have just arrived in town, settled your next days excursion and eager to go see some real Patagonia. At night, grab a chair and go outside. The non polluted sky shows stars like you will never see again, and don´t be surprised by satellites passing by!
Excursions to Moreno glacier (sliding scale entry fee to National Park, 100 pesos for entranjeros, 40 for people of south america and 15 for argentinos), by bus or there are many agencies in town offering private trips with ice walking driking whiskey with glariar ice etc....
There are three very expensive bus companies (120 pesos round trip - 90 minutes each way), leaving around 8:00am in the morning or so and one company (Cal Tur) also at 13:45. The return time is around 16:00pm. The bus will drop you off at the port for boat trips and take the rest onward to the glacier before coming back for you. Try to sit on the left side of the bus for the best views of the glacier and surrounding landscape.
More conveniently a taxi is 300 pesos round trip, so if there are at least three of you its easy maths.
There is a hideously expensive (by any standard) restaurant onsite (meagre coffee in styrofoam cup is 12 pesos), be a cheapskate and take a picnic and flask and keep smiling. Numerous places to sit and eat overlooking the glacier.
The Hielo y Aventura Minitrekking excursion - morning trip to the observation boardwalks, boat ride across the lake, and then an hour and a half with (provided) crampons walking around on the Glacier. Proper foul weather gear (rain jacket and pants) and trekking shoes/boots are strongly recommended, and can be rented from several places in town. Bring sunscreen, and don't bother with water (you can drink the water off the glacier).
The Fernandez Campbell All Glaciers boat tour. All day long, but some incredible glacier views that you won't be able to see otherwise. When booking, see if you can get on the smaller boats (they have one large and three small ones).
Visit a ranch. Ask your travel agency where to go. Recommendations ; El Galpon del Glaciar Ranch, Nibepo Aike Ranch, Rincón Ranch, Alta Vista Ranch. On most estancias you can ride a horse.
A must is "La Toldería", a fast food that turns into a disco after 1AM. The pub of El Calafate.
Food is expensive mainly due to the remote location of the city. There is a supermarket at the end of the main street with fair prices. Buy your food for your tours in advance, because on ruta 40 and in the glacier national park there are almost no places to buy food.
Handicraft and souvenirs can be bought in large scale at the main street.
Argentina´s wine is one of the finest. Take advantage of the exchange rate and enjoy yourself with them. Or drink whatever you want, like beer from all over the world, wiskeys, liquors, champagne, etc. Or a Diet Coke :) Pomelo sodas are something different.
Be aware that in order to cross the border to chile, you have to ask the car rental company to arrange you some documents.