As a starter system, I would recommend getting a 275 gallon IBC tote or something similar and raise 75 tilapia in it. Attach at least 275 gallons worth of grow beds (roughly 48 cu ft of grow beds) and just do a syphon setup and constantly pump the water from the fish tank to the grow beds and back (or look into chift pist basically adds a sump tank so that only 1 water pump is needed). It's a large enough system to get a real world test out of it, and small enough that it can be done inexpensively.
Prawns (large shrimp) can co-exist with tilapia in tanks kept above 55 degrees.
In cooler climates, rainbow trout can be grown in an aquaponics system. They might need a bit more oxygen, since they are used to living in streams.
Bass have done well in fish-farming situations and might do well in aquaponic tanks.
Although not considered an edible fish, Koi a good ornamental fish for aquaponics. They are a colorful, long-living, adaptable fish and produce lots of beneficial waste nutrients for grow beds.
Barramundi are one of the most popular aquaponics fish, especially in Australia. They are a warm water species and are noted for their feisty disposition and savory flavor. Barramundi prefer water temperatures which stay in the low 80's.