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The obvious places you truly cannot miss: The Blue Mosque and Hague Sophia are amazing, the main bazaar is a fun walk through, walking through Taksim in the evening is a kick, and walking along the Boshperus is great - although I personally wasn't overly impressed with the boat ride up and down it. However, I don't think any of these will be particularly newsworthy to you, I have gathered from my own looking that you could do a 1 day trip to Ephesus if you so chose, otherwise, I am interested in hearing the replies, as have pretty much the same question.
-TBri

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Is there anything you should know?Yes,Be aware that amazing Istanbul can steal your heart away!
To go to your question on safety for a huge seething city of some 14 million people Istanbul is remarkably safe far more so than many other comparable places.Crime against visitors is usually petty theft and pick pocketing nothing more,and even this could be cut down if visitors were a little more careful with their personal belongings.
This thread may be of interest to you,mainly aimed at solo travellers but it still has useful info for you-
http://www.lonelyplanet.com/thorntree/thread.jspa?threadID=2072508

From the things you say the Galata/Beyoğlu area could be a good district to stay in.Sultanahmet the old city offers a matchless old skyline but it is touristy and limited for evening entertainment.Beyoğlu too is an old district but it puts you amongst great shopping and nightlife opportunites and getting around to other neighbourhoods is so easy via tram or ferry-or feet of course,Istanbul is an easy city to discover on foot!

Maybe go and experience a real Turkish bath,hamam and be pampered.Learn about Turkish cooking,there are several excellent half day cooking courses on offer.Or learn about traditional Turkish hand crafts at a short course doing something like marbling (ebru)or calligraphy.Depends what you like to do.
Istanbul is a foodies heaven and for a good night out with good food too head one evening to a Meyhane,a traditional Istanbul drinking house.These are mainly found around the Asmalımesçit area of Beyoğlu.
http://kozano.wordpress.com/2008/08/07/meyhane-etiquette/

All of the ancient must see's are located close together in Sultanahmet,Blue Mosque,Aya Sofya,Cisterns,Topkapı Palace,Archeological Museum,Arasta bazaar,Grand Bazaar etc.but also head up the Golden Horn to Fener and Balat and visit the Kariye Museum,(Chora Church) too.
It will be a bit chilly to do a full Bosphorus cruise but there are regular 90 minute cruises to give you a taste and see the city from the water.Just wait until a dry bright day then go and jump on board.Tur-Yol boats leave from Eminönü pier every hour or so for this and it costs around 12TL.
Hard to know what else to suggest without knowing your interests or how many days you have there,but I hope this helps :)

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sarikanarya: thank you so much for all of that information.

i definitely wanted to check out a turkish bath at some point. can you recommend a good one, not touristy.

anything else you can think of would be greatly appreciated.

thanks again.

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Cemberlitaş and Çağaloğlu hamams are famous and in beautiful historical buildings but they are very touristy and a bit too much conveyor belt-ish for me.Sulimaniye hamam is nice,less touristy but in a lovely complex.It is also safe and respectable for ladies too:)
http://www.suleymaniyehamami.com.tr/index.asp
There is also this one but you need to have an über big budget to go here!
http://www.visit2istanbul.com/haseki-hurrem-sultan-hamam-bath/

For another nice little mooching head to Çukurcuma district with it's curio,antique and vintage shops,a nice area to stroll.
Head up the Golden Horn on the bus through Balat and Fener to Eyup.This is a nice few hours visit taking you through some of the very old districts and the former Jewish quarter of the city,you'll also find Chora Church(Kariye Museum )here.
Just across the Golden Horn from there you'll find the Rahmi Koç Transport Museum which is very good.
Also try to get over to the Asian side for a few hours,Kadıköy is a nice regular district a world away from the tourism hot spots,quite arty and laid back with lots of little lanes and side streets to mooch around and you can take the nostalgic tram to Moda an older part of the neighbourhood.The 20 minutes ferry trip offers great photo opportunities too.

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As with many other capital cities Istanbul has a red double-decker hop-on hop-off bus that costs $25USD for a day. Suggest you spend your first morning on that bus to get oriented. The bus passes most of the tourist sites - the round trip takes 90 minutes and starts at Sultanamhet Square ( although with a ticket you can get on of off at other designated stops).

Just google "Istanbul double decker bus" for more details.

Edited by: chris1953

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This blog page about Istanbul may give you more information.

A tour company posted this itinerary of the double decker bus.