Quick answer: The best time to visit Istanbul is April–May or September–October. Expect a mid-range budget of about €65 per day. Visa: Visa-free or e-visa for most passports — check your nationality.
Why this window
Istanbul spans two continents; cluster by shore. Sultanahmet's icons, Karaköy's cafés, a Bosphorus ferry and the Asian side make four dense but walkable days.