Airport transportation is a bit easier from the New Town (north of the Golden Horn) rather than the Old City (south of the Golden Horn). But the unmissable sights--even though each of us would have a different list--are in the Old City. For ease of airport access, I'd recommend staying in the New City and taking a taxi, but for ease of sightseeing I'd recommend the Old Town. On my recent very long trip I initially stayed in the Old City and knocked out just about all the sights in that area. Then after visiting Bursa I stayed in the New City and saw things on that side of the Golden Horn and took a couple of boat trips. On a short trip like yours, there is no single, right answer.
The atmosphere is very different in much of the Old City; it would be a severe culture shock to some people. Have you been to Naples or Palermo? Much of the Old City is more crowded (also more "foreign") than those places.
Some of the Rick Steves tours stay at the Hotel Acra, in the Old City east of the Blue Mosque. I don't know anything about the hotel itself (may not even have walked past it), but I found myself in that area one afternoon and remember thinking it was considerably less crazy-hectic that the area west of Hagia Sophia and the Blue Mosque. There are a bunch of hotels in that area as well as many restaurants. Yes, it seemed quite touristy in the sense that there weren't thousands of local residents out on the sidewalks, but I think many folks would be happier there than in the area west of the two most popular mosques. It is a longer walk to the bazaars and to the tram line (easiest way to get across to the New City), but it's a good location for the Basilica Cistern, Hagia Sophia, the Blue Mosque, Topkapi Palace and the Archaeological Museum. For airport transportation, this doesn't look very convenient; the distance would be somewhat longer. However, the taxi route might be able to skirt the worst of the Old City traffic congestion; I just don't know.
Warning: A few blocks east of that hotel area is Kennedy Caddesi, the main drag along the coast (possibly useful for airport taxis). Do not start walking north along the landward side of that road. The next place to turn back inland will be way north, and the sidewalk is mostly narrow and occasionally broken up, so you won't cover the distance quickly. In a few places the sidewalk disappears completely and you have to step into the road to move forward. Traffic moves quite fast. If you want to walk or jog in that area, cross Kennedy Caddesi to the water side as soon as you reach that road. You cannot cross the road anywhere unless you want to vault over the median--not impossible for a young person, I'm sure, but watch the traffic!
Hiring a driver for a day would be useless in central Istanbul as it would in any major European city. Istanbul traffic is extremely congested, so it's easier to walk short distances unless there's a mobility issue. There are lots of taxis around, and I'm sure the official rates are low. A taxi shouldn't be expensive even if it does get hung up in traffic--if it charges you the official rate (i.e., if it uses the meter correctly). I cannot tell you the odds of that happening because I used public transportation.
Get the City Mapper app (or use Google Maps) so you can check on whether there's convenient public transportation that will get you from where you are to where you want to be. Unfortunately, that means either buying individual transit tickets or buying an IstanbulKart stored-value card. The vending machines can be tricky, though some work fine and some tourists have no problem with them. I had massive problems until I found a machine in better working order and fed it currency rather than trying to use my credit card. The machines do take US credit cards. Ask your hotel where you can buy an IstanbulKart. It will come with TL 30 loaded, enough for two rides (no free transfers).