Depending on your budget my recommendations are the following areas :

1. Nisantasi/Tesvikiye : This is generally an upscale residential area with good shopping and restaurants including some quite reasonably priced ones. There is not much public transport except "dolmus" , a shared taxi type large van to three destinations from different points. However, with a large family, the cost of a taxi will not be higher than public transport anyway.

2. Sisli/Pangalti/Kurtulus : Close to Nisantasi, rather similar but more middle class. Sisli is on the Metro line. but Pangalti and Kurtulus possibly more family friendly depending on the street.

3. Cihangir : A bit bohemian, many expats. getting to be expensive. Stay in area between Akyol-Susam-Akarsu-Sira Selviler streets for the better neighborhoods. For example the very popular Witt suites are on a steep hill and difficult to handle with a toddler.

4. Ayaspasa/Gumussuyu : We have lived in this area all our lives and love it, but some parts have steep hills and the only restaurants and cafes are on the main street. If apartment is close to Gumussuyui Caddesi, you will be close to the bus lines and restaurants as well as the transportation possibilities of Taksim square.

5. Besiktas ; this one has some rather old building stock but has good shopping and market streets for the modern middle middle class with a great deal of eating options at great prices. It is a large area, but most have good access to either the coastal Bosphorus road with its ferry boat landing piers or Barbaros Blvd, the main route leading down to that road. The problem here is that there usually are not too many touristic rental apartments available.

The only company I know and have used for friends and in-laws coming from the States is "Cheya Residences". They and we were quite satisfied with them.

I strongly recommend that you do not stay in any neighborhood in the old city, unless you are a devout moslem.

Suggested day trips are :

Yalova for spa and thermal baths.
Iznik (Nicea) for Byzantine ruins, architecture, Ottoman tiles, natural beauty, lake promenade.
Sile-Agva, Black Sea views, natural beauty, swimming
Princes' Islands : horse carriage rides, walks, nice homes, natural beauty, donkey rides for the kids.
Bosphorus cruises.

Kids will love the Rahmi Koc Museum.

Other attractions for the kids will be Florya aquarium, Miniaturk, Rumeli Hisar fortress, The city walls, some of the historic sites like the cisterns, Galata Tower