Welcome to the forum Armgarwal! You say you have made some reservations;does this include your hotel in Istanbul?I am guessing so but...if you haven't then consider splitting your time there to 2days at the start and 2 at the end rather than returning there just to depart.
To get from Antalya to Cappadocia you have 3 options;
a) by bus,this can be overnight or day bus but o/n would be better given your limited time.
b) hiring a car which you could keep to use around Cappadocia and hand back as you leave.Do you plan to fly back to Istanbul?If so you can drop off the car at the airport as you leave.
c)flying via Istanbul.This can be both expensive and long-winded if connections are not good.
For any flights not yet booked try to book evening flights to keep your daytimes as free as possible rather than eat into your daytime sight seeing hours.
Kaleiçi is a nice base for Antalya as it has lot's of character to it,you can spend ages wandering around in there :)The castle and marina are close by and you can walk along the seafront and beach to the far end where the excellent Antalya Museum is located.