Istanbul local bus system

I don’t know anything about Istanbul, but I see that CityMapper covers it, and I use CityMapper religiously wherever I go (and where I live), so that is what I would try.

I don't recall seeing buses in Istanbul, but there is a tram system we used several times. I googled several sites for maps. Also found taxis to be reasonable.

I was there in 2011, so things could certainly have changed since then. But at that time, the tram, funiculars, and boats were all sufficient to get me anywhere I wanted to go - if just walking was not sufficient, which it often was. I remember that getting to the Chora Church required a bus (I didn't see this sight), but for most other places of tourist interest, the buses were not needed.

If I did want to go somewhere in Istanbul now, I'd start with Google Maps. Many (but not all) local transit systems are loaded into it, so it often has excellent directions.

If you're trying to get from the airport to Taksim square, the Havas shuttle buses will take you there. Just confirm which one at the airport. There should also be a fare machine where you can load a pre-paid transit card (I think it's called Istanbulkart or something like that) to pay for the bus. Yes, there are tons of buses in Istanbul - perfectly nice, modern. but crowded buses with articulated stops (not very different than the ones in Seattle). If you're worried about getting lost or not knowing where your bus stop is, perhaps you can just use the maps function on your phone and follow the arrow to see where you are in real time. I took a bus north and south along the Bosphorus, but it was very straightforward and I knew my way around and where to get off because I studied maps beforehand.

In at least some views of the D53 results at Maçka Oteli stop, Google Maps mentions "shared taxi" and I think it's based on a late-night search time. You might be better off starting with "Maçka" when searching on the IETT bus site, to let it suggest stop names, and also to choose a realistic time of day on either Google or other web sites.

There are tons of buses and many, many bus routes in Istanbul. Buses, trams, metro (underground subway), and express buses are all over Istanbul. The IETT map shows all the public transit that is run by the city. The D53 is a Dolmus (pronounced Dole moosh) which is a shared taxi or minibus. Dolmus and mini buses are privately run, not run by the city of Istanbul, that is why they are not on the IETT maps.
MooveIt seems to have some Dolmus/minibus routes, but not all. GoogleMaps is the same. I don’t remember if the privately run Dolmus and Minibuses accept the IstanbulKart (pre paid loadable public transit fare card) or if you need to pay with cash. All city run public transit (including ferries) accept the IstanbulKart, no cash.
To get to the Chora Church (Kariye Muze) , definitely take a taxi. It’s complicated to get there any other way. Do have Google Maps turned on so you can make sure where you are going. If you ask a taxi driver to take you to the Chora Church, he will have no idea where you want to go. You must call it by the Turkish name of Kariye Muze (recently switched to Kariye Mosque).

Thank you all for the helpful information. Happy travel.