A lot of people have difficulty finding a successful strategy the first time they play Fish Tycoon. It's very important to take care of your starting fish. If they die you may not have enough money to buy any more eggs, and you'll have to start a new game.
To take care of the fish you get at the beginning of the game, make sure to feed them only when they're hungry (check the "Fed Meter" at the bottom right of your screen, below the "Health Meter"). If you feed them before they're hungry, they will lose health. During testing, I actually killed fish by just feeding them a lot!

Also, if a fish gets sick, it will lose health and (eventually) die if you don't heal it. If you don't have enough money to buy medicine, you should either trash the fish or separate it from your other fish, since disease can spread.
There are many different strategies to get a good start in Fish Tycoon, but here is one way that will definitely work in all versions of the game:
Drop the starter eggs into Tank 1. Go to the supplies area and buy some common eggs for $160. Drop them into Tank 1 and wait for them to hatch. If you want them to grow up quickly, you can set the game speed to fast, but you'll have to check your fish often to make sure that they are fed and healthy. They will grow up in less than 2 hours.
Once they are adults, try to find two fish EXACTLY the same. You can always breed them successfully, and their babies will not lose health and die. If you can't find two fish exactly alike, choose two fish that are very similar (the same type of fish - for example Spotanus or Beta). If you breed these, even though their fins might be different, they will probably live to be adults. You will usually get at least a few fish similar enough to breed successfully, but, if not, you can always start a new game and try again. There are some hybrid fish that will live until at least adulthood and bring good prices, but be prepared to "lose" some baby and teen fish during the search for those higher-priced fishies. If you take care of your starting fish, you can always keep breeding and trying different combinations
Once your new fish are grown up, you can either sell them or use them to breed more fish (keeping in mind that many hybrids will die until you have better Fish Environment).
Keep breeding fish as described above until you have enough money to buy a second tank. Tank 2 will give you more room to breed fish and you will make money faster. Drag some of your fish to Tank 2 and keep breeding the same fish to make enough money to research tank improvements.
Remember to keep making money by breeding fish and selling them. Once you have made enough money to research Fish Environment, you will be able to successfully breed and keep more expensive species of fish.