You basically need to keep track of the fish combinations that you're trying, or you'll tend to keep breeding the same fish over and over. Assuming that you have upgraded your tanks to the highest level of Fish Environment, you should be able to breed any fish combination without them dying. One method, if you don't want to bother with tracking things, is to breed one fish with every other fish type that you have and keep the new fish that you find. Then breed those offspring with the original fish. You should soon find fish you've never seen before. Once you've done that for a few generations, you can repeat the process with a different set of fishies.
If you want spoiler-type solutions, check out the Spoilers threads in the
Fish Tycoon Guide section of the forums.