There are a few fishing shops in the Kemeralta. Not the entrance by the clock tower, the part near Kızlarağası Hanı, a large old pink stone building. You will find the shops in the street that runs alongside it.
I know quite a lot of people fish along the cordon in Izmir but I wouldn't fancy eating anything that came out of there as the water is very polluted.
There are plenty of good places withing easy reach of Izmir where the water is cleaner although I don't know how much luck you will have due to some idiots who have used dynamite and therefore killed off all the living things in the vicinity.