I go to Istanbul quite a lot, and always stay in Kadıköy now. There are lots of small hotels and they are generally a bit cheaper than on the Asian side, but never listed in any of the guidebooks. Kadıköy doesn't have any major attractions, but I think it is a nice more relaxed area to stay than Sultanahmet or Beyoğlu. The area around the market (on a map, look for the streets around the Greek and Armenian churches) is probably the best place to stay, as there are many restaurants, bars and cafes, and quite a few old buıldings have been turned into hotels. On the other side of the main busy street, there are also a lot of hotels, but street noise can be a problem in some.
It is quite far to get to the old city and Beyoğlu/Taksim, but I can't think of a better way to start the day than taking a ferry across the Bosphorus. Ferries leave every 20 minutes or so over to Eminönü (for the old city), Karaköy (for Beyoğlu) and less regularly to Beşiktaş (for Dolmabahçe Palace), and there's also a metro station, taking you over to the European side with one change. Public transport runs until quite late (the ferries stop at around 10 or 11), so unless you plan on a late night on the other side, getting back to Kadıköy won't be too difficult - there are also minibuses called dolmuş which run from Taksim once the other options stop.
For the airport, there are regular buses direct to both the new İstanbul Airport on the European side and Sabiha Gökçen Airport on the Asian side.
I also recommend going to the Princes Islands (Adalar) for a day trip if you want a break from İstanbul's traffic. Again there are ferries from Kadıköy to all four main islands.
Wherever you stay in İstanbul, get an İstanbulkart from any bus, metro or ferry ticket office and often kiosks nearby, which can be used on most forms of transport.
Hope you enjoy İstanbul - it's one of my favourite cities!