Many people going to Aconcagua hire a US-based outfitter or guide service to manage their climb. This is probably easier, and avoids a lot of complications and issues.
Argentinian Spanish is a slight bit different from other Latin American dialects, especially Mexican Spanish. English is not extremely common among local speakers in the Aconcagua region.
As a Latin American country, sometimes the clock is set a little behind. Sometimes by a day or more. Many of the things we have come to expect to happen quickly just won't.
There is quite a bit of going around in circles to accomplish somewhat simple tasks. Sometimes money can fix this, sometimes not.
A guide service connected to a local logistics provider can cut through a lot of this, or hide it behind steak dinners and gear checks.
That being said, let's proceed for those who want to go it alone.
First of all, you have to get to Mendoza Argentina. This can be daunting. Yes, just go online and book American Airlines through Santiago Chile. You'll end up on LAN Chile Airlines, and there've been rumors of odd baggage transfers on this flight combination. Or you could book on Delta through Buenos Aires, but then you end up with an airport transfer and on Aerolinas Airlines for a leg, and it's a bit vague about the 15kg luggage restrictions.
Or you could book from LAX or MIA to MDZ on LAN Chile through Santiago, and just use whatever your favorite US carrier is to get to the starting point, but then you'll have to manually transfer your luggage to LAN. Oh, well.
And to top it off, Chile has odd food import rules and it's a possibility you'll have all your food confiscated (though I've heard rumors about yummy snacks being taken by Argentinian Customs agents as well). Also, if you leave the terminal you might have to pay a reciprocation fee of $150 in Chile, and sometimes you have to leave the terminal for the transfer between airlines.
Some people like to just get off in Santiago and take a bus to Argentina (only a few hours, really, on a road that is truly survivable, from what I hear) but the border crossing again, is risky for your gear and food (not to mention the reciprocity fee at the airport). This bus might not actually go straight on to Mendoza, and you have to go to Mendoza first to pick up a permit in person, even though the bus passes right by the trailhead to Aconcagua. Oh, well again.
Anyway, now, one way or another, you've managed to get to Mendoza. You have to stop by a government office to pick up an invoice for a permit. This office follows government type business hours with a little bit of laxity to allow for it being a tourist industry after all. The taxi drivers know where it is. Take the invoice to a bank down the street and pay. Get a receipt. Take the receipt back to the government office and get your permit. Permit fees are based on dates between November and March for High, Mid and Low season. You can get a trekking permit up the trail to base camp and back, or a climbing permit. Both have expiration dates. It is remotely possible to climb outside that little calendar window, but you will be completely alone, and it's very serious stuff. You've been warned.
While you're in Mendoza you'll need to get your supplies too. Many people recommend taking a taxi to Walmart where you can get just about everything you would normally need. I've heard they have gas cylinders like MSR and Jetboil use, but can't confirm that myself. They also have white gas, though you can get that elsewhere (paint stores oddly enough).
From here you'll need to get yourself and your stuff to Penitentes or Puenta del Incas, both little towns near the trailheads and where the mule companies take off. You'll need to give them your stuff in plenty of time for them to weigh it (your stuff will need to be split into two even loads less than 30 kg ea) and ensure that the mules will be loaded and ready to go in time to beat you to the basecamp (assuming the normal route).
Okay, mules? Yes. I've read stories of some real hardcore go-it-alone people doing it without mules. For many people this is a three week expedition. Carrying that much stuff on your back sucks, and this isn't Alaska, so sleds are out.
The mules can do the normal route in a day. It will take you at least two. They will beat you, so keep that in mind when you drop off your stuff.
Also, be aware that a guided group will normally allow as many as 3 to 4 days for all of this up to the point where you give up your stuff and start up the trail. It would be very difficult to do it all in one day, with a tight schedule worked around flights, government offices, banks, shopping, driving, and who knows what all else. The buses to these little towns have set schedules and sometimes require reservations the day before. If you have an evening arrival you'll have to wait for the next day to get your permit and the next day to get on the bus, then the next day to drop off your gear. Four days right there. If you're having "cultural experiences" and steak and wine dinners, and street markets, you don't really notice all that so obviously. (Mendoza Argentina has some world famous steak restaurants and wineries)
From the trailhead you will encounter a ranger who will verify your permit, that it is correct for your intended adventure, and the correct season. Your time starts now. You will be issued a numbered garbage bag and you will be fined stiffly if you lose it. On the normal route you will spend the day hiking to Confluencia where you will camp. In a few books there are other camps mentioned, but according to a few local sources, you will camp in Confluencia. If you come from sea level, it might be a good idea to hang out here and hike a few surrounding canyons or peaks to acclimatize for a day or two before proceeding to Plaza de Mulas, the base camp for the normal route at over 14,000'. Hiking from the trailhead to base camp can take from two to four days.
At the Plaza de Mulas base camp you may encounter another ranger for a permit check. I've heard of medical checks as well. Most people would need to hang out here at 14,000' and just get used to the altitude slowly, the recommended protocol to prevent AMS (acute mountain sickness).
That's it for now, the basics of how to get there. I will correct any of these that are wrong, though for each of them I've read several trip reports and blog entries that I feel support my suppositions.