Bus service within Istanbul is frequent, and drivers and riders tend to be helpful, so you should be able to navigate your way to major tourist stops like Eminönü, Taksim, and Beşiktaş. You must have an İstanbulkart to board, and the fare is 1.95 TL.
For travel around the country, Turkey has an extensive system of intercity buses, and Istanbul's large, chaotic Esenler Otogar is the heart of it. Esenler itself is a bit out of the way, though easily accessible by metro from Aksaray. Alternatively, most of the major bus companies have offices located near the top of İnönü Caddesi (which winds down from Taksim Square to Kabataş), from which they operate shuttle buses, known as a servis. These take passengers either to Esenler or to their own ministations on the main freeway, allowing you to avoid making the trek out to Esenler via public transport.
A second, smaller bus station, at Harem on the waterfront on the Asian side of the Bosphorus, is easily accessible by ferry from Eminönü.