How much TL should we get at an ATM?

There have been a few threads in this topic. The general consensus has been to get as little as possible in cash since the value is falling so rapidly and to try to do as much as possible with your credit card.

I’m not sure what that minimum amount is, but hopefully someone with recent experience will post.

I wrote a trip report from my Best of Turkey tour in May/June:
https://community.ricksteves.com/travel-forum/trip-reports/3-weeks-to-fall-in-love-with-turkey

If you search that thread for "lira" you'll find all sorts of hints about ATMs and the things that I found I needed cash for (e.g., small souvenirs, occasional cash for toilets at rest stops, tips for hotel staff, drinks on the boat rides in Istanbul and Antalya)

The big ticket items for which you may need cash are the optional boat ride and the optional Turkish Bath, both in Antalya. We were able to pay credit card for the optional balloon ride.

are credit cards well accepted for admission to most sites?
Yes, I don't recall any site that didn't accept credit cards.

How about a nicer restaurant meal?
Yes, most restaurants and many cafes accepted credit cards. However, tips are expected to be paid in cash and would be frowned upon to try to add them to the bill. Most street food kiosks are cash only.

I've been trolling this forum for ideas on sites and activities not covered on the RS tour.
That same trip report has ideas of things that we did on our extra days in Istanbul, including Dolmabahçe Palace and the Museum of Great Palace Mosaics, which is near the Blue Mosque and Hagia Sophia.

More details on all of this in the trip report. And feel free to ask any questions there (I'll get an email notification) or here (I won't get an email notification, but lots of travelers will reply.)

Even the Havaist bus ticket from IST Airport to Taksim Square can be paid for with a credit card for about $3.50 per passenger. It makes a few other stops as well. As others have said— get just a few days cash at a time since some expect at some point the Turkish Lira could crash in value overnight. The Lira I had from my visit 3 years ago cost about 7 per US $1. Last June $1 bought 17 Lira.
Also, there is a great deal of variation among the bank ATM’s and the commissions they charge and the rate they impose to get those Lira. I found one mainline Turkish bank—unfortunately I don’t recall the name— that did not rake off 15% to 18% between the service charge and the exchange rate when getting Lira from the ATM as several did. Be sure to carefully read the conditions of the transaction at the ATM.

I usually get 1,000 TL -- about $56 -- when I go to an ATM. I have had pretty good luck with Halkbank ATM's --- both for fees and ease of use. If I run out of money, I just hit the ATM again.

Tourist and high end restaurants usually take credit cards, but you can expect to pay European prices. Locandas, pide joints and little hole-in-the-wall places have great food at much more reasonable prices .... but I don't expect them to take credit cards. I was shocked at the difference in restaurant prices between Beyazit and Sultanahmet.

If you are taking private transportation from the airport to your hotel, try to book the transfer through your hotel. (Our hotel simply added the transfer cost to the bill.) Taxis can be huge ripoffs in Istanbul. If you are taking public transportation, check to see if the rail line is open to the airport. It was still under construction when I was there in November, but it may be open now.