Around Sultanahmet (historical peninsula, touristic area) you can pay with USD in stores and restaurants. However, museums will not accept foreign currency. You can either pay with Turkish Lira (TL) or credit card.
There are ATM's all over the city and you can withdraw money from your credit or debit card.
In the Grand Bazaar, almost all currencies are accepted and so are the credit cards (all but American Express! Since Amex charges the stores high commissions, it is usually not accepted or preferred by many stores).
Taxis may except USD, however it would be handier to have TL.