Unless you have money to buy new eggs, you won't be able to get more fish without starting a new game. If you're playing either the Mac or Windows version, you can do that by going to the menu and clicking on Change Player. Then you can either delete the game and start a new one, or start another game in a different slot.
This thread has links to posts that explain why fish die in Fish Tycoon and provide some non-spoiler tips for getting started successfully.
