It's pretty easy to fly to Irkutsk these days. It's a major city in a major country, and Irkutsk airport handles dozens of flights each day.
It's even easy to book flights to Irkutsk - as with anywhere in the world you can now book flights to Irkutsk online with absolutely no hassle - it's the same as booking a flight back in the United States.
The most common way to fly to Irkutsk is to fly from one of the two Moscow airports. A flight takes about five hours (it does go halfway across Russia!) and most flights are run by Aeroflot, the Russian national carrier and Transaero (Russia's second largest airline). This does mean that, if you are coming from outside of Russia, you will probably also need to get a flight from your home country across to Moscow, Russia. But if you book with someone like Aeroflot, you can actually buy one ticket that covers both your flights.
You can fly direct from a few countries outside of Russia, but not many. Because of Irkutsk's close links with Asia, there are regular flights to Beijing, Mongolia and Seoul, Korea, as well as to many Central Asian countries like Tajikistan and Kazakhstan. The only European country that you can fly direct to Irkutsk from though is Germany, because an airline there runs charter flights from Munich.
If you're afraid of flying, there are many other ways to get to Irkutsk. The most common is to take the train. Irkutsk is a major stopping off point on the Trans-Siberian railway, and you can get to Irkutsk easily either starting from Beijing in China, or starting from Moscow or St Petersburg in Western Russia.
You can get to Irkutsk by other routes - you can drive, motorcycle or, if you're mad enough, cycle on a pushbike (some people really have done this!), but I wouldn't recommend it. If you've got any sense, you'll either fly to Irkutsk or take the train, then you'll feel nice and relaxed by the time you make it to the warm bedroom of the hotel in Irkutsk where you'll be staying.