Getting from the airport to Karaköy:
Cheap way: Take the airport shuttle from either airport to Taksim. Once you are there look for the metro sign (white, navy, red M letter) and go down.
You will first have to purchase an Istanbulkart (our version of the London oyster card). See how it is done here: http://www.turkeytravelplanner.com/go/Istanbul/Transport/istanbulkart.html Note that a fare costs 2TL so you should stock up accordingly, but one fare will also get you transfers. The vendors that dispense and fill the card are also in English, however if you have difficulties with figuring out how to get the card or to get it filled, do not hesitate to ask people around you. As a rule, Istanbulites are very friendly to visitors, and will be happy to help.
Once you have your Istanbulkart take the funicular from Taksim to Kabatas. From Kabatas take the tram to Karakoy. You will know you are in Karakoy when you see the flat bridge that spans the Golden Horn. Get off before the bridge. But again, you can also ask someone on the tram to tell you where to get off.
If you are coming in from Ataurk airport, there is a second way to get to Karakoy: Take the metro from the airport. (Again, you will need to get an Istanbulkart). Change from the metro to the tram at Aksaray, which is the end of the line, and then transfer to and continue on the tram to Karakoy. (If you take this route, please be aware that you will be in Karakoy after you cross the flat bridge that spans the Golden Horn, whereas coming from Kabatas you will be getting off before).
A map for metro and tramlines can be seen here.
Fancy way: You can take a taxi from the airport to Karakoy. Coming from Ataturk airport (the main airport in Istanbul) this will cost you around 50TL (17Euros), however if you are landing at Sabiha Gokcen it will be considerably more. I cannot give an exact price since I have never done this, but my guess would be around 130-150TL (40-50Euros) since this is actually a very long ride.