I would say three days is more than enough in Istanbul but probably just 2 will be fine. Turkey is a big country so you really need to sleep else where. Plus staying in a town overnight is part of the experience.
Anyway after Istanbul I would suggest spending a day in Bursa which is a beautiful Ottoman town just south of Istanbul.
Then heading down the coast go to Galopoli to see the WWI battlefields and the memorials commemorating the 250,000 British, Turkish and ANZAC troops that died on the Gallipoli Peninsula. It's all seeable in a day
Then close by is Ephesus which is the ruins of a roman city which is spectacular. Selcuk the nearest town is also quite nice but again not worth spending too much time in.
Heading further south you get to the Mediterranean coast which is relaxing or intense depending on where you go.
Towns like Bodrum have the biggest clubs and serve exclusively english food and everyone there is English. It wasn't my cup of tea but if your looking for a party bodrums the place to go. Another similar place is Antalya (which is slightly nicer).
However the south coast has many places to relax too. One of my favourite place in Turkey is Olympos near Antalya which is a small collection of hostels in tree houses in a valley and a beautiful beach. It's got a really nice atmosphere and most people spend much longer then they expect there.
However my favourite of all places is Cappadocia in the middle of turkey. It's a little far to go but its well worth it.
Goreme in Cappadocia is the village where most people stay and is where all of the houses are built into the strange rock formations... i.e.
http://www.tatilsepeti.com/Files/TurResi…
I spent a long time there riding round the mountains on a hired moped and getting drunk on the local spirit raki.
If your a little more adventurous then you could head east to mount nemrot which is apparantly amazing,and further to kurdistan but travelling there can be a little difficult at times.
There are probably more places worth mentioning but I forget. Guide books like the lonely planet usually have recommended itinerary for 10 day trips so check them out.
But remember don't try and see to much. Turkish buses are notoriously stuffy and can be cramped so traveling every day can be extremely tiring.