Lots of good advice directly above from acraven. I'll just add a couple more bits, and respectfully offer a more nuanced view on one subject: taxis.
Before jumping in a cab and zooming off, always negotiate the fare (you will be amazed at how many taxis in Istanbul have non-working meters...). Hold up a calculator/phone to display numbers, be clear about where you want to go, agree to a number and stick to it. Don't expect much/any English language skills.
Istanbul has some of the worst traffic on Earth. Large sections of the city are often gridlocked, not moving much or at all for long times. A taxi to/from the airport may be a good option (that's what we opted for last time, it kind of worked out...) but hailing a cab on the street to go short or even medium distances within the central city can be a bad choice. We were trying to essentially cross the old city center, going from one side of it to the opposite. We were tired, and had limited time, so we flagged down a taxi, well, actually 4 taxis eventually. The first one just drove away when we asked them how much it would cost to take us to our destination. The next two gave us a price that seemed ridiculous, we tried to bargain for a more reasonable fare but those cabbies would have none of that, they just said no and drove away. Finally we got a guy who was wiling to haggle and agreed to what seemed to us to be a reasonable fare (in Turkish Lira, the equivalent of around 10-15 Euros). The agreed-to fare was higher than it should be, but we were on cab #4 and losing patience. The driver was visibly unhappy, fretting verbally about the price he had agreed to and apparently about...the traffic. We jumped in, drove about 100 meters, and were immediately engulfed by gridlock. And there we sat, stuck, occasionally crawling a few meters but never actually moving very far, for a long, long time. The driver was not happy. He was chain smoking the whole time, too. After about 20 minutes of this, we had gone just a few blocks, and were actually now further away from our destination than we were when we got in the cab. We had had enough, realized we had made a bad choice, and figured it was time to cut our losses. I handed the driver the amount we had initially agreed to, made hand gestures to indicate we were giving up, he nodded, I opened the door and we stepped out in the middle of a large totally gridlocked street. and walked to our destination.
Point being: Even if you are up for the taxi adventure, the traffic can make opting for a cab a poor choice. It depends on the neighborhood, time of day, and more. We had much better experiences with the extensive and (mostly) easy-to-use public transit system. Yes, getting the transit card initially can be a confusing puzzle, (we eventually figured it out).
A couple minor suggestions: If you are referring to Turkish Airlines, it's "code" is TK (not TA - here, many people would read "TA" as "TripAdvisor" another popular travel forum).
Important: The "free stop-over option" that TK offers is not a general, free-for-all, anything-goes, here's-a-free-hotel-night for everyone. Your flight needs to meet some very specific criteria to qualify. You should look at their criteria very carefully and make sure your flights clearly meet them. I am a fan of Turkish Airlines (and the country) and I've never used this benefit myself. FWIW I've not heard before of you being able to choose how many nights you plan to stay there - my understanding is that, if you qualify, you get one free night in a hotel, and the next day you continue your onward flights, with no negotiation about staying extra nights. I have not followed the qualification criteria closely so I could be wrong about that (but I'm not going there expecting a free hotel room). You will want to take a hard look at the rules before making plans.
Good luck, have fun - Istanbul is amazing.
