best neighborhoods to stay Istanbul?

Most of the top historic attractions of Istanbul—Topkapi Palace, Hagia Sofia, the Blue Mosque, the National Archaeological Museum, the Grand Bazaar— are located in the district of Sultanahmet (“Sultan City”). Just across the Galata Bridge over the Golden Horn is the district of Beyoglu which includes Taksim square and a number of lively residential neighborhoods such as Cihangir, Karakoy and Kabatas. Attractions in Beyoglu include the Galata Tower, the palace of the last Sultans Dolmabahce Palace, the famous pedestrianized shopping street of Iskitial Caddesi, the Karakoy Fish market and a number of museums.

While Sultanahmet is close to many of Istanbul’s top attractions, many say staying in Beyoglu, with a population of 233,000, allows you to see more Turks than tourists with more exposure to Turkish culture as few Turks ( fewer than 1,000) actually live in Sultanahmet which consists mostly of monuments, mosques, and government buildings. Consequently, prices at restaurants are much more expensive in Sultanahmet as restaurants and stores cater to international travelers.
You can reach Sultanahmet from Beyoglu by metro, tram and ferry boats in about 30 minutes. You can also reach it by walking over the Galata Bridge, which everyone should do at least once while in Istanbul for the panoramic views the bridge offers of one of the most dramatic urban settings in the world.
While Istanbul has plenty to hold my interest for a couple weeks. it is an exciting, intense urban experience that might best be broken up after 5 days at the beginning of a trip by going to Selcuk for a couple days to visit Ephesus, and then maybe 3 nights in Goreme for a visit to Cappadocia, followed by a final 5 nights in Istanbul to top off your trip before flying home.
You can fly from Istanbul ( Sabiha Gokcen (SAW) Airport is most convenient as it’s the smaller of Istanbul’s two airports) to Izmir’s (ADB) Airport and then taxi to Selcuk in 45 minutes or take the train in 60 minutes. Trains from Izmir’s Airport depart every 4 hours for Selcuk.
For Cappadocia flying SAW Airport to tiny Kayseri Airport ( ASR) and take a shuttle to Goreme in 40 minutes.
Fares and flight schedules can be seen at www.Skyscanner.com

I stayed in Beyoglu, about between Taxim and Galata tower, and thought it was a great location for it’s local-ness with asset to sights/monuments.

I stayed in Sultanahmet but that was 16 years ago but if I went back, I would stay in Beyoglu. I enjoyed Sultanahmet, especially the proximity to all the sights, but I never ate there as restaurants prices were high and the food was so-so. And back then, I'm sure it wasn't as touristed as it is now. My hotel (which was £40 a night back then) actually had a car pick me up (along with another couple from Australia) at the airport - I doubt you'll find that service today without paying premium prices. It was a nice hotel, too - they had a balcony where you could eat breakfast with a beautiful view of the sea. But as I said, I think Beyoglu would be a better bet.