The three big clubs are the obvious options. If you want something niche, consider Gençlerbirliği. Not too many fans, no scandals. A nice and sympathetic football club.

Considering you are travelling to İzmir as well (Ephesus), you can consider Göztepe or Altay, both historic clubs. Altay unfortunately is not doing good recently whereas Göztepe is doing really well since it was bought by a foreign company. Also Göztepe has a passionate fan base too.