Easiest way from the Prague airport to Prague. Shuttle! Bus? Train?

I plan on using a taxi, unless someone convinces me otherwise. My flight home leaves at 6 am so no other option for that anyway. The website for Vaclav Havel airport gives info on buses and trains, also taxis--I've heard AAA is a reputable one.

Easiest way is to have your hotel book a fixed price car service. Our driver met us on exiting with our baggage. In our case we had ski box with 210 cm long skis and the hotel arranged for a van that would accommodate the ski box.

I'm sure that there are cheaper ways, but easiest is a car service.

For cheap, public bus is cheap:
http://www.prg.aero/en/parking-transport/transport/public-transit/
Bus Fare Note:

The fare for a piece of luggage, a dog not in a carrier or a baby
carriage without a child is 16 CZK. The time and zone validity of a
ticket for The fare for a piece of luggage, a dog not in a carrier or
a baby carriage without a child is the same as the time and zone
validity of the ticket of the passenger transporting the luggage or
dog, up to a limit of 300 min.

If you're considering public bus #119, the info on the web link above is probably correct, but note that the metro stop served is a recent change from info you may see in guidebooks. We learned it here: https://community.ricksteves.com/travel-forum/czech/changes-to-metro-system-in-prague. See also http://www.huffingtonpost.com/huff-wires/20150410/eu--czech-transport-trouble/.

I found the AE Airport Express bus to be convenient to my hotel on the last trip, since it ends near Namesti Republicky.

http://www.praguetransport.com/
this is a highly reliable company I have used many times, run by an American who has made Prague his home for over 20 years.all the driver speak English and are very informative.cost is $26 dollars each way
Laura mentions changes to the 119 bus and I will be experiencing these changes in 4 weeks time. if you want to use public transport from the airport to your hotel let me know where you are staying and I can advice.
The AE bus that Laura mentions is great for certain areas of the city but has limited times especially in the morning and late evening.The AE bus stops at namesti Republicky and final stop in at Hlavni the main train station and is run by the train company.
If you have an early morning or late evening return to the airport book a shuttle such as the one mentioned or sort things out with your accommodation.

...AE bus stops at namesti Republicky....

If I recall correctly there is a taxi stand near by and Náměstí Republiky is between Old Town and New Town. It's only about a enjoyable km walk to the Clock and Old Town area.

Namesti Republiky is an ideal area for any first time visitor to Prague and an area I always seem to spend some time in when I am in the city. Good transport links to various parts of the city, a big shopping mall, some good pubs and resturants , the magnificent Municiple building and easy access to all the main sites.there is a nice farmers / craft market on there a couple of times a week too.The fair price taxi stand is beside the Palladium shopping mall.

As Edgar says the easiest way is to get transportation through your hotel if they do it. Otherwise taxi or private car company as Unclegus suggests. The cheapest is public transport. Then AE which goes to the Main Train Station stopping on the way at namesti Republiky (square of the Republic). Namesti Republicky means square of the Little Republic.