Probably would be able to see a lot of the city in 4 to 5 days max.
Blue mosque- should be on everyone's list to see. Get there early to avoid the crowds but avoid pray times and remember to dress appropriate.
Topkapi Palace - is really good. Pay the extra for the ticket into the harem. The audio tour doesn't include alot extra that isn't written on the signs in English around the site. Also there are lots of English speaking guide taking groups around so just keep your ears open to what they are saying.
Haghia Sophia - was great in the early morning due to less people. The upper level is beautiful Mosaics and should be on your must see list.
Basilica Cistern - very interesting and worth a look. Take the time to set up your camera and get some great shots in the low light.
Archaeology museum- was a good to see if you have the time but not an essential Istanbul visit. The history is Istanbul section was good. Also interesting Troy section.
Sogukcesme Sokagi- cute street to walk down and look at the wooden buildings and little art gallery courtyard. Doesn't take much time and connects the Topkapi palace back to the metro tram line.
Golden Horn- catch the ferry from small pier behind the university near the Galatea bridge, there are signs at the small port indicating Golden Horn ferry. It departs once an hour at 45 mins past. Costs 2TL. Ferry zigzags it was up the golden horn terminating at Eyup. I think this is a highlight of Istanbul.
Pierre Loti Cafe - this is reached by either a 10 min cable ride or a 25min walk up the ramp and stairs through an old ottoman cemetery. Catch the cable car up in my opinion and then walk down via the old ottoman cemetery taking the time to look at the headstone. The cafe itself has fanatic view over the golden horn and this combined with the ferry ride is a must do if you can find the time for a a half day out.
Gulhane Park- close to Topkapi palace. Was a nice walk through but no real highlights.
Galata bridge- strolling over this bridge is a must. Take time to look at the fisherman and views over the water bustling with boat traffic.
Museum of Turkish and Islamic arts - worth a look if you time. Nice display of Turkish rugs and examples of wooden inlay. Great cup of tea from the cafe and good views over the blue mosque.
Hippodrome - will only take 5 to 20 mins to see and it is just outside the blue mosque and try to imagine the chariots racing around.
Tomb of Sultan Mahmut II - have a quick look of you have time. Was closed when I went for a visit.
Constantines Column - has interesting engravings and if you walk past stop for a look as it is in a centralish spot.
SS Sergius and Bacchus Church - the little haghia Sophia is a short walk from the blue mosque. Look for the orthodox influence in this now mosque.
Spice bazaar - worth a look but better prices can be found in the street around the market and heading towards the grand bazaar.
Valens Aquaduct- a nice walk up to the aqueducts if you have time, cross over the street and visit the Kadinlar Pazari pedestrian square to find some good Kurdish restaurants serving Buryan kebap.
Vefa Bozacisi - this little shop serves boza. A drink made from millet. A flavor like nothing I have tired and worth the walk to taste this unique food.
Suleymaniye mosque - worth a look due to size and design. Take time to explore the surrounding grounds.
Princes mosque - have a look if walking in the area but wouldn't go out of your way to visit.
Tulip mosque - easy to visit as near the tram line and interesting design. If you have the time take a look.
Grand bazaar - is massive and very interesting to walk around in. A must see but good shopping can be found in the small street between it and the spice bazaar.
Mevlevi lodge - take the time to stop in at this museum when you are walking down Istiklal Caddesi. Good to learn the history behind the whirling dervishes.
Whirling dervishes show - take the time to see a show if you can. Interesting performance but most are very touristy shows.
Galata tower - on a nice day go up and admire the view over Istanbul. Good thing to do towards the end of your visit so you can look at all the place you have visited.
Istanbul museum of modern art - something to see on a hot day when you want to get out if the heat or a raining day to stay dry but not a must see. The photographic gallery has a nice display of Turkish pictures.
Taksium square - this is the place to go for a night out. Lots of younger people with bars and restaurants in the area.
Church of St Stephen of the Bulgars - closed for renovations but from what I could see through the fence would have been nice to have a closer look at the cast iron structure.
Church of Pammakaristos - this is slightly difficult to find but worth the effort if you feel like a walk up hills. The mosaics are nice and the outside of the building is beautiful. Don't need to spend very long there and would rake 3rd for mosaics after Aya Sophia and St Saviour.
Theodosian walls- if you have the time they are interesting to see for the large and layered construction that is still imposing today.
Yedikule fortress - this open air fortress museum requires you to catch the train to yedikule and then a short walk to the fortess which is alright to see. Nice view over the sea of mamara from the towers but NOT a must see.
Church of St Saviour in Chora - this church is away from main tourist sites of the old city but the detailed mosaics inside are worth the trek. This visit will take 40 mins depending on how many pictures and details you want to look at.
Ciragan Palace - this is on the shore of the bosphorus and can only been seen from the water as the building itself is closed off as a very classy hotel with sightseers not welcome.
Yildiz park - this large garden is nice and the place for local to take their wedding photos. There are 3 large ottoman Pavilions within the grounds that house a cafe and also the sale pavilion operates as a museum you can look inside at the 18th century royal European interiors but not really worth it. The park overall isn't that special.
Yildiz Palace - this is located outside the park. Do not try and enter from the park by the same name as you will encounter a large mote and wall despite what it may look like on a map. The inside of the palace is a small museum with some military displays and royal household items. Was very ordinary and is a long steep walk uphill to reach it.
Dolmabahce Palace - be prepared for the possibility of a long wait for tickets. Maybe best to try early morning. Only guided tours offered, not allowed to just walk around yourself. There is the harem and selamlik ( if only doing one then this is the best option) choices. Both tours will take 2 hours to complete once inside the grounds.
Princes island - this is a great day trip out of Istanbul. Catch the ferry ( 4 or 5 TL) from Kabatas to any of the four largest islands in the group. The largest island Buyukada is the last stop. hiring a bike and riding around for 2 to 3 hours to see most of island is super fun. The walk up the st geroge monastery is steep but nice views over the island and back to Istanbul. Great day out and try to fit it in if you have enough time on a sunny weekday so not as busy.
ortakoy sunday market - very average. The products local like cheap fake imports.
Istanbul eating food walking tour- this was great day out trying all different foods from small local places. If you have the time and a very empty stomach this is something you should try to fit in as it is very enjoyable day.
Cooking class at cooking alaturka- this restaurant holds class twice a day which last around 4 hours including time to sit down and enjoy your meal. Was an alright class and learnt a couple of new recipes.
Iskender kebap- this flat kebap is my favorite and there are a couple of chain store in istanbul from the original one in Bursa.
Turkish delight ( lokum to the locals)- varying prices from 7TL to 70 TL a kilo can be seen. In the Streets around the spice bazaar you are more likely to spot a local buying there than inside the bazaar itself due to the prices. Don't be afraid to ask to try some before you buy. I like the rose, pomegranate and saffron flavors but it is really up too you to decide your favorite :)
Halva- this is a sweet slice/ fudge made from sesame and nuts. Taste good.
Kaymak- this is a clotted cream / philly like products that is worth a try. Grab some with honey and sweet bread roll for a little bit of heaven in the from of cream.