I'll be coming from the airport so I won't have Turkish lira on hand unless I try to change some money at the airport.
Never "change" money (anywhere, ever, IMHO). It's easy and cheap to pull cash (in Turkish Lira) from an ATM at the airport. Use your Debit/ATM card, linked to your account at home. This is always the cheapest way to get local cash. Just alert your bank/CU in advance, and find out if they charge any fees. The "money changers" always charge fees, usually quite high.
The airport is huge - actually, vast (one of the biggest in the world, and feels like it) - and modern. There are ATMs everywhere. You will be doing some walking there - it'll give you a chance to stretch your legs after the flight.
Is it uncommon to pay with credit card?
No, but don't count on it. You can always ask - and if you hope to do so, ask before you jump in the cab. Also check to see if they are using a meter - they should be but for some strange reasons a lot of the cabs have meters that appear to be turned off, because they are "not working"...In that case, get an estimate from the driver, or choose another cab (you may find that if you try to find another cab with a working meter, the "broken" one in the first cab may miraculously start working...). Be prepared for shenanigans but don't worry too much, keep your wits about you.
The airport is far from the city. Expect a cab to cost in the neighborhood of US$60-80, possibly more - depends on traffic and your destination (that estimate is from 2 years ago, it may have changed since - Turkey has spectacular inflation, with big swings in the exchange rate). Of course, pay in Turkish Lira, not Euros or dollars (some people will ask for payment in Euros, since its more stable; you will probably get a poor exchange rate, so always pay in local currency unless you can not). As a rich foreigner looking for a cab, you should be able to pick and choose, so don't be bullied into paying too much or no meter or other nonsense.
Some cabs (not all) will accept payment by credit card (much more likely at the airport than a cab hailed on the street). You can shop around until you find one who says that's OK (but don't be afraid of getting cash from airport ATMs, you will need cash soon enough, might as well get some at the airport - just use the same common sense precautions about sketchy-looking ATMs that you would at home). ATMs inside the airport should be fine. They are all over the place.
Presumably you know about the scourge of Dynamic Currency Conversion?