There are probably as many different strategies as there are people playing the game, but the most important thing is to make sure that you always have at least several common fish in your tanks (or a batch of common eggs) until you have better Fish Environment, and to care for them so that they stay healthy. Otherwise, you could find yourself with nothing but a bunch of fragile hybrids that won't survive and no money to buy anything with. Look for breeds that do well in the starting tank environment and sell for a reasonable price (at least $20) and keep breeding those fish to make money.

For one guaranteed approach to getting started successfully, you can take a look at this post. It will not be the fastest road to riches, but it will allow you to get a good start in the game.