Bebek is a neighborhood, so above all I like places there that tell me a story, that give me an impression, have some sense of continuity.
If you are a morning person, you could jog walk or just stroll through the Bebek Türkan Sabancı Park, ending up near the sea, by the small and quaint "Bebek İskelesi" and breathe in the salty morning air coming your way...
Go on walking towards the Egyptian Consulate admiring its art nouveau architecture. Then, you could cross the street, and maybe visit 4 Mevsim Simple Life, a lovely home & personal accessories shop (http://www.simplelifehome.com) just slightly further up towards Arnavutköy. Even if you are not in the mood for shopping, I am sure you could hardly resist the temptation there...
Ok, walking towards Rumeli Hisar, maybe now it is time you stopped for tea and simit at Bebek Kahve right next to the Bebek Mosque, by the sea. Or if you prefer Italian coffee, just walk in Cafe Nero in the center of town and help yourself to the morning coffee of your choice.
Right across the street from Nero, I love the Bebek Badem Ezmesi (founded 1904), with its perfectly delicious pistachio and almond marzipans.
Again along the same block is my favorite bookstore Türkü, and its owner Nuran Hanım is usually there for a chat even if you don't buy anything. At Türkü, you will find more than an interesting selection of books both in Turkish and English: silver Ottoman design jewellery, scarfs in best quality Bursa silk, old engravings, examples of Ottoman caligraphy and many more curiosity items.
Further down the street is Yargici, a very good Turkish retail shop for women, and across the street is its accessories branch.
Well, you could easily miss Mini Dondurma, a really tiny ice cream shop as its name suggests, founded in the 60's, selling the best ice cream in town. Once you try their ice cream, you'll know what I mean.
For lunch and brunches (and happy hours) my favorite spot is Mangerie or the Bebek Hotel Bar is also a landmark although seems to have lost its popularity to to Lucca, a real yuppie joint. (Not my cup of tea.)
If it is tea time, it would be lovely to stop by at Divan and have sip your tea practically "on" the water.
For dinner, or a special night out, Bebek Balıkçı is really remarkable with its catch of the day and delicious mezes... as long as you're prepared for the bill.
Last but not the least, is Baylan Pastanesi, another classic. Although quite a new in the neighborhood, it was founded back in 1923...
Even if you don't feel like eating, drinking or shopping, just walking along through Bebek to Rumeli Hisar is an exciting enough itinerary, watching huge ships, tiny boats, ferries as well as sea gulls and cormorants on the Bosporus...and watch out for people fishing with their huge fishing rods.