My husband and I just returned from a 2 week trip to Vietnam that included a free stopover in Istanbul on the way back. We took Turkish Air to Vietnam and the airline offers free stopovers in Istanbul and I highly recommend taking advantage of this generous program. In fact, my 4 previous visits to Istanbul all occurred as stopovers on my way to or back from somewhere else. This was my husband's first time and he wanted to prolong the experience so we opted for a 6 day stopover. I detest connecting flights as they are too stressful for me. If the first flight gets delayed, you run the risk of missing your next flight whereas a long layover, especially if you're allowed to leave the airport, makes things much better for me so taking advantage is a no brainer.
Personally, I like the New Town part of Istanbul better as a base than the Old Town. Yes, the Old Town is for all the biggies like the Blue Mosque, Hagia Sofia, Topkapi Palace, Grand Bazaar and the Spice Market, The Mosque of Suleyman the Magnificent, etc... However, because the Old Town has all the biggest tourist attractions, it also has the most tourists and those who prey on tourists. I find the prices of food and drink to be higher on this side and find shopping isn't as straightforward as in other parts of the city. Also, the Old Town will present to you many handsome Turkish men who will flatter you, ask where you're from, and invariably try to get you to buy a carpet, come into their store, restaurant, or any other such thing as to sell you something. This doesn't happen in the New Town.
We stayed in the New Town, very close to the Galata Tower. I like this area as it's very lively at night whereas the Old Town gets quiet after 6pm when the tourist attractions close down. In the New Town, you're likely to get few to no aggressive sales pitches and more transparency in pricing. It's just a different vibe.
Public Transit in Istanbul is among the best in the world. It's easy to use, is safe, fast, and clean and will take you wherever you want or need to go. With the Istanbul Kart, sold at yellow kiosks in or near all the public transit stations, instructions come in many languages, and it's easy to use and top off. For pennies a ride, you can get anywhere in the city.
My top tourist attraction in the city is the Mosque of Suleyman the Magnificent. It rivals the Blue Mosque and Hagia Sofia but is better because it gets far less tourists and the views of the city from the grounds are truly stunning. Of course, you must check out the Blue Mosque because it is incredible, but unfortunately, it's so overwhelmed by tourists, it'll be hard to take in the grandeur, but at least it's free so no risk of losing a large chunk of money to get in. Hagia Sofia is not worth the cost and time to get in. It used to be before it was turned back into a mosque from a museum. Back then, it was one of the most magnificent man made sites in the world and worthy of all the accolades heaped upon it. Now that it's a mosque again, visitor areas are sharply reduced which is strange as visitors are allowed in all the other mosques in the city as long as it's not during services, but for some reason, the first floor of Hagia Sofia is off limits to all non muslim visitors. Also, it's covered inside and out by scaffolding and because it's a mosque again, some of the Christian iconography is blocked. The worst part is the ticket sellers will induce you to buy the very expensive combo ticket that includes admittance to the museum for about $60 USD per person when all you need is the ticket to the actual Hagia Sofia. Still expensive and still disappointing compared to how it used to be, but at least you'll only be out $30 USD.
I could go on and on about my trip, but I see I'm getting close to my maximum length and will be cut off soon. If you go, and I hope you do, Rick's Istanbul book is a godsend and I promise will make your stay as good as it can get. Do all the walks.