Step by step guide to get started successfully:
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. They will grow up in less than 2 hours.
Once they are adults, try to find two fish EXACTLY the same. You can breed them successfully. 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. If you don't get any fish similar enough to breed successfully, you can always start a new game and try again. It's important not to cross breed species early in the game, because they will die and you won't have any money to buy more eggs.
If any of your fish get sick early in the game, you should probably not spend money on medicine until you have extra money. Once your new fish are grown up, you should move them to the selling tank (Tank $), and click the Sell Fish button. You will see the store screen. When a customer buys one of your fish the species, picture and price of that fish will be displayed in the information area below the store.
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 breed more expensive species of fish.