If you want to get the golden fishing rod then you will need a lot of patience. Here is how you do it.
1 - Get yourself a fishing rod
2 - Catch at least one of every variety of fish in the game.
Very simple in theory, very time consuming in practice. Fish appear at different times of day and different months of the year so it is very hard to know that you have all of the fish. But it is also a good way to earn some bells to pay down your mortgage so in the end it is definitely worth the effort to catch every fish that you can.
I personally have never been able to get the golden fishing rod, but I aspire to it someday.