Istanbul has a fully integrated public transportation system that can be used by the same payment method: Istanbulkart. Istanbulkart is an electronic card that you can buy, charge and use to get on public transportation. You can provide the card from newspaper stands, subway, tram or ferry stations. You need a pay a down payment of 6 TL to get the card and charge it to use. You can charge the card from here, ticket machines, automatic charging stations, or from the app (Android, iOS). You can buy the card from street sellers too but they are likely to give you an overpriced, uncharged card. My suggestion is, to always use the "biletmatik" machines located on the stations but be careful, the machine will take 6 TL for the card and charge the rest of the money to the card, you cannot get a refund so use smaller bills. The alternative to the Istanbulkart is jetons, but even with the card price, if you will take more than 3 rides, Istanbulkart is cheaper. Istanbulkart can be used by multiple people and will give you discounts on transfers.
Moving on to public transportation choices.
Subway and railway systems
Istanbul has a very wide railway system that you can use to reach almost everywhere including passage from the Bosphorus via Marmaray. You can find the maps here. You can check out how to go from point A to B from here but my suggestion would be using Google Maps which is fully integrated with the whole network. Unfortunately, neither airports are accessible via the subway. You can take the world's second-oldest subway between Karaköy-Beyoğlu (Istiklal Street).
Bus and Metrobus
Busses in Istanbul have different colors. Greens are environmentally friendly, purples and yellows are disabled accessible and turquoise ones are owned by private companies. You can check out which busses to take from here which is also integrated to the whole system or you can use Google Maps.
Metrobus is a bus that has a private lane on the roads which means you don't get caught in traffic. You can find more info from here.
Havaist's are the buses that work between most spots in the city and the new Istanbul Airport. They are the most accessible way to get to the or from the airport. The schedule can be found here.
Minibusses can be seen all over the city which can only be ridden by cash. You can check out the routes from here.
Ferry and Seabus
The ferry is a very pleasant way of getting from one point to another and accessible from most spots. You can enjoy the beautiful view while sipping your tea and get away from daily troubles for a quick 15-minute ride. For timetables and prices, you can check out here. You can also see Bosphorus tours from the same page.