There are two ways to get from Taksim to Sultanahmet. They are:

By public transport:

If you are in Taksim square, take the metro from Taksim to Kabatas and you can get to Sultanahmet from Kabatas by tram. If you are at the other side of Istiklal take the little metro (calles as “tunel”). It takes you to Karakoy. You can take the tram from Karakoy to Zeytinburnu direction and get off when it arrives to Sultanahmet. These won’t take much time. However, there is an alternative way;

By taxi:

You can take a taxi, that will cost higher. It’s the best way to use public transport during commute hours as the traffic is very crowded at that time.

Getting From Taksim To Sultanahmet is a post from: Travel Istanbul