Walking tours:
Museums and such: Haghia Sophia, then on to the Topkapi museum (these two should take at least three to five hours), preferably along the road in the back of the Haghia Sophia, where there are some nicely restored houses. Then on to the Blue Mosque and the square with the obelisks on it (At Meydani). Along its side is the very good Museum of Islam Art. Descend slightly and find the small Haghia Sophia with its nice garden (it was under restoration, but you probably can get in). Then uphill to the Sokollu Mehmet mosque complex, top notch tiles inside.
Take a tram or walk to Eminönü (where the boats leave for trips to Asia or up the Bosphorus). Visit the New Mosque at the back, then the Egyptian Bazaar next to it, and going further in that direction, locate the Rüstem Pasha mosque with its excellent tiles. It's on a raised platform near an old clothes market, you may have to ask directions. Then take a cab or find a bus to Eyüp mosque complex, a mile or three up the Golden Horn. Visit this Eyüp complex at your leisure (the mosque is not particular, the court is, and the milling of believers, with many boys-to-be-circumcised amongst it; a Friday might be a good day to do this). Then, if you have the stamina, it might be nice to walk back too; maybe all the way (five miles or so), but taking a route along part of the city wall to first the famous Kariye Church with its mosaics, then on to Selimiye Mosque with its great view on the Golden Horn (and a fine mosque by itself), then the Fatih Mosque (passing through some very religious and lively neighborhoods), then on to the well-restored Sehzade mosque, and next to Süleymaniye (don't forget to enjoy the view from the Golden Horn side). If you have some energy left, you might go on to the University complex, and by then you are very close to the Beyazit mosque. A book market (it's small) is behind this good, unexceptional (nice courtyard though) mosque.
Once again go to Eminönü, but this time take the boat (those large ferries) to Üsküdar. You will arrive before a fine mosque in front, another one four hundred meters off to the right, slightly inland behind a traffic roundabout, and a third, very small, at the sea front. See the market stretching inland, walk about and don't forget to walk along the shore, maybe eating a fish meal in one of the bobbing boats along it. This is a good visit for late afternoon, early evening, fleeing the city. You will be joined by thousands of people going home from "town" but the way back will be on a near-empty ferry. The frequency of ferries will go down in the evening, so make sure there is a connection back.
Go to the railroad station and find a Sirkeci-Halkali suburban train, and get out at (from memory, Yedikule station). You will be quite close to Yedikule, a nice fortress, and will have fine views of the city walls. The trains leave every 15 minutes or so, the ride is peculiar (the material is bad, but if you are in luck every second stop another salesman will enter and try selling his wares, it's fun). The ride is takes anywhere from twenty minutes to half an hour. This is not a "must", but it can be great fun.
You will have missed the covered bazaar in all this. That is because you will get there anyhow. If you go to Beyazit and the book market you are almost at two of its many entrances. Try and find the Nuruosmaniye Mosque and its complex at the other side, it's worth it. And after having explored the covered part, take a relaxing walk downhill, into the general direction of Eminönü, where it is "uncovered bazaar" all the way. Cross the Galata bridge to see some things on the Northern side (for instance take the "tünel" teleferik ride up much of the hill (entrance close to the opposite side of Galata bridge, ask around)), then continue to Taksim. Shops are of the international variety.

Beyoglu: is the heart of Istanbul's nightlife. Situated across the Golden Horn from Sultanahmet, it is a great place to watch street performers, find excellent coffee and party.

Sultanahmet: is the historic heart of Istanbul, featuring many of the city's most famous attractions, including the Topkapi Palace, Hagia Sophia, the Blue Mosque and more.

Ortakoy: is undoubtedly one of the most beautiful parts of Istanbul, hosting the Ortakoy mosque right under Bosphorus bridge.

Kadikoy: is the commercial centre of the Asian side suburbs.