In Istanbul's medieval core, traffic is heavy and one-way streets make routes circuitous, so walking is often the best way to go distances up to one km (0.6 mile).
For example, the 2-km (1.2-mile, 25-minute) walk from Sultanahmet along Divan Yolu to Beyazit Square and the Grand Bazaar takes you past many interesting old buildings and shops. You could travel this route via tram, but then you wouldn't get to stop and visit the sights along the way.
The walk downhill from the Grand Bazaar along Uzunçarsi Caddesi through the Tahtakale district to the Egyptian (Spice) Market in Eminönü is one of my favorites, an utterly un-touristy tour through the heart of Istanbul's local crafts and shopping district.
You can take a taxi from Sultanahmet to Taksim Square, but it's more fun to take the tram downhill to Eminönü, stroll across Galata Bridge, then take the Tünelunderground train uphill from Karaköy (Galata) to TünelSquare at the southwestern end of Istiklal Caddesi, formerly the Grande Rue de Péra.
Stroll northeastward along the avenue toward Taksim Square. If you're tired or in a hurry, you can take the restored 19th-century tram from Tünel via Galatasarayto Taksim, but lots of little side streets invite your exploration along the way, and you've got to be walking to see them.